An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term. For example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6. An arithmetic sequence can be known as an arithmetic progression.
Contents
 1 How do you find the arithmetic sequence?
 2 What are the 5 examples of arithmetic sequence?
 3 What is Z in arithmetic sequence?
 4 What is arithmetic sequence give example?
 5 How do you determine the common difference of an arithmetic sequence?
 6 Arithmetic Sequences and Sums
 7 Arithmetic Sequence
 8 Advanced Topic: Summing an Arithmetic Series
 9 Footnote: Why Does the Formula Work?
 10 Introduction to Arithmetic Progressions
 11 Finite Sequence: Definition & Examples – Video & Lesson Transcript
 12 Examples
 13 Finding Patterns
 14 Finding Sums of Finite Arithmetic Series – Sequences and Series (Algebra 2)
 15 Arithmetic Sequence – Formula, Meaning, Examples
 16 What is an Arithmetic Sequence?
 17 Nth Term of Arithmetic Sequence Formula
 18 Sum of Arithmetic sequence Formula
 19 Arithmetic Sequence Formulas
 20 Difference Between Arithmetic and Geometric Sequence
 21 Solved Examples on Arithmetic Sequence
 22 FAQs on Arithmetic sequence
 22.1 What are Arithmetic Sequence Formulas?
 22.2 How to Find An Arithmetic Sequence?
 22.3 What is the n thterm of an Arithmetic Sequence Formula?
 22.4 What is the Sum of an Arithmetic Sequence Formula?
 22.5 What is the Formula to Find the Common Difference of Arithmetic sequence?
 22.6 How to Find n in Arithmetic sequence?
 22.7 How To Find the First Term in Arithmetic sequence?
 22.8 What is the Difference Between Arithmetic Sequence and Arithmetic Series?
 22.9 What are the Types of Sequences?
 22.10 What are the Applications of Arithmetic Sequence?
 22.11 How to Find the n thTerm in Arithmetic Sequence?
 22.12 How to Find the Sum of n Terms of Arithmetic Sequence?
 23 Arithmetic Sequence: Formula & Definition – Video & Lesson Transcript
 24 Finding the Terms
 25 Finding then th Term
 26 Arithmetic progression – Wikipedia
 27 Sum
 28 Product
 29 Standard deviation
 30 Intersections
 31 History
 32 See also
 33 References
 34 External links
 35 Arithmetic Sequences
 36 What is an arithmetic sequence? + Example
 37 13.2: Arithmetic Sequences
 38 Finding the Number of Terms in a Finite Arithmetic Sequence
 39 Solving Application Problems with Arithmetic Sequences
 40 Arithmetic Sequences and Series
 41 Arithmetic Series
How do you find the arithmetic sequence?
An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence then subtract it by the previous one, and the result is always the same or constant then it is an arithmetic sequence.
What are the 5 examples of arithmetic sequence?
= 3, 6, 9, 12,15,. A few more examples of an arithmetic sequence are: 5, 8, 11, 14, 80, 75, 70, 65, 60,
What is Z in arithmetic sequence?
Jun 24, 2018. Assuming r is the constant difference between two consecutive terms, you express z=y+r in terms of y and z=x+2r in terms of x.
What is arithmetic sequence give example?
What is an arithmetic sequence? An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term. For example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6. An arithmetic sequence can be known as an arithmetic progression.
How do you determine the common difference of an arithmetic sequence?
An arithmetic sequence adds or subtracts a fixed amount (the common difference) to get the next term in the sequence. If you know you have an arithmetic sequence, subtract the first term from the second term to find the common difference.
Arithmetic Sequences and Sums
A sequence is a collection of items (typically numbers) that are arranged in a specific order. Each number in the sequence is referred to as aterm (or “element” or “member” in certain cases); for additional information, see Sequences and Series.
Arithmetic Sequence
An Arithmetic Sequence is characterized by the fact that the difference between one term and the next is a constant. In other words, we just increase the value by the same amount each time. endlessly.
Example:
1, 4, 7, 10, 13, 16, 19, 22, and 25 are the numbers 1 through 25. Each number in this series has a threedigit gap between them. Each time the pattern is repeated, the last number is increased by three, as seen below: As a general rule, we could write an arithmetic series along the lines of
 There are two words: Ais the first term, and dis is the difference between the two terms (sometimes known as the “common difference”).
Example: (continued)
1, 4, 7, 10, 13, 16, 19, 22, and 25 are the numbers 1 through 25. Has:
 In this equation, A = 1 represents the first term, while d = 3 represents the “common difference” between terms.
And this is what we get:
Rule
It is possible to define an Arithmetic Sequence as a rule:x n= a + d(n1) (We use “n1” since it is not used in the first term of the sequence).
Example: Write a rule, and calculate the 9th term, for this Arithmetic Sequence:
3, 8, 13, 18, 23, 28, 33, and 38 are the numbers three, eight, thirteen, and eighteen. Each number in this sequence has a fivepoint gap between them. The values ofaanddare as follows:
 A = 3 (the first term)
 D = 5 (the “common difference”)
 A = 3 (the first term).
Making use of the Arithmetic Sequencerule, we can see that_xn= a + d(n1)= 3 + 5(n1)= 3 + 3 + 5n 5 = 5n 2 xn= a + d(n1) = 3 + 3 + 3 + 5n n= 3 + 3 + 3 As a result, the ninth term is:x 9= 5 9 2= 43 Is that what you’re saying? Take a look for yourself! Arithmetic Sequences (also known as Arithmetic Progressions (A.P.’s)) are a type of arithmetic progression.
Advanced Topic: Summing an Arithmetic Series
To summarize the terms of this arithmetic sequence:a + (a+d) + (a+2d) + (a+3d) + (a+4d) + (a+5d) + (a+6d) + (a+7d) + (a+8d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + ( make use of the following formula: What exactly is that amusing symbol? It is referred to as The Sigma Notation is a type of notation that is used to represent a sigma function. Additionally, the starting and finishing values are displayed below and above it: “Sum upnwherengoes from 1 to 4,” the text states. 10 is the correct answer.
Example: Add up the first 10 terms of the arithmetic sequence:
The values ofa,dandnare as follows:
 In this equation, A = 1 represents the first term, d = 3 represents the “common difference” between terms, and n = 10 represents the number of terms to add up.
As a result, the equation becomes:= 5(2+93) = 5(29) = 145 Check it out yourself: why don’t you sum up all of the phrases and see whether it comes out to 145?
Footnote: Why Does the Formula Work?
Let’s take a look at why the formula works because we’ll be employing an unusual “technique” that’s worth understanding. First, we’ll refer to the entire total as “S”: S = a + (a + d) +. + (a + (n2)d) +(a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n1)d) + After that, rewrite S in the opposite order: S = (a + (n1)d)+ (a + (n2)d)+. +(a + d)+a. +(a + d)+a. +(a + d)+a. Now, term by phrase, add these two together:
S  =  a  +  (a+d)  +  .  +  (a + (n2)d)  +  (a + (n1)d) 
S  =  (a + (n1)d)  +  (a + (n2)d)  +  .  +  (a + d)  +  a 
2S  =  (2a + (n1)d)  +  (2a + (n1)d)  +  .  +  (2a + (n1)d)  +  (2a + (n1)d) 
Each and every term is the same!
Furthermore, there are “n” of them. 2S = n (2a + (n1)d) = n (2a + (n1)d) Now, we can simply divide by two to obtain the following result: The function S = (n/2) (2a + (n1)d) is defined as This is the formula we’ve come up with:
Introduction to Arithmetic Progressions
Generally speaking, a progression is a sequence or series of numbers in which the numbers are organized in a certain order so that the relationship between the succeeding terms of a series or sequence remains constant. It is feasible to find the n thterm of a series by following a set of steps. There are three different types of progressions in mathematics:
 Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP) are all types of progression.
AP, also known as Arithmetic Sequence, is a sequence or series of integers in which the common difference between two subsequent numbers in the series is always the same. As an illustration: Series 1: 1, 3, 5, 7, 9, and 11. Every pair of successive integers in this sequence has a common difference of 2, which is always true. Series 2: numbers 28, 25, 22, 19, 16, 13,. There is no common difference between any two successive numbers in this series; instead, there is a strict 3 between any two consecutive numbers.
Terminology and Representation
 Common difference, d = a 2– a 1= a 3– a 2=. = a n– a n – 1
 A n= n thterm of Arithmetic Progression
 S n= Sum of first n elements in the series
 A n= n
General Form of an AP
Given that ais treated as a first term anddis treated as a common difference, the N thterm of the AP may be calculated using the following formula: As a result, using the abovementioned procedure to compute the n terms of an AP, the general form of the AP is as follows: Example: The 35th term in the sequence 5, 11, 17, 23,. is to be found. Solution: When looking at the given series,a = 5, d = a 2– a 1= 11 – 5 = 6, and n = 35 are the values. As a result, we must use the following equations to figure out the 35th term: n= a + (n – 1)da n= 5 + (35 – 1) x 6a n= 5 + 34 x 6a n= 209 As a result, the number 209 represents the 35th term.
Sum of n Terms of Arithmetic Progression
The arithmetic progression sum is calculated using the formula S n= (n/2)
Derivation of the Formula
Allowing ‘l’ to signify the n thterm of the series and S n to represent the sun of the first n terms of the series a, (a+d), (a+2d),., a+(n1)d S n = a 1 plus a 2 plus a 3 plus .a n1 plus a n S n= a + (a + d) + (a + 2d) +. + (l – 2d) + (l – d) + l. + (l – 2d) + (l – d) + l. + (l – 2d) + (l – d) + l. (1) When we write the series in reverse order, we obtain S n= l + (l – d) + (l – 2d) +. + (a + 2d) + (a + d) + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a … (2) Adding equations (1) and (2) results in equation (2).
+ (a + l) + (a + l) + (a + l) +.
(3) As a result, the formula for calculating the sum of a series is S n= (n/2)(a + l), where an is the first term of the series, l is the last term of the series, and n is the number of terms in the series.
d S n= (n/2)(a + a + (n – 1)d)(a + a + (n – 1)d) S n= (n/2)(2a + (n – 1) x d)(n/2)(2a + (n – 1) x d) Observation: The successive terms in an Arithmetic Progression can alternatively be written as a3d, a2d, ad, a, a+d, a+2d, a+3d, and so on.
Sample Problems on Arithmetic Progressions
Allowing ‘l’ to signify the n thterm of the series and S n to represent the sun of the first n terms of the series a, (a+d), (a+2d),., a+(n1)d S n= a 1 plus a 2 plus a 3 plus .a n1 plus a n + (l – 2d)+ (l – d)+ l.+ (l – 2d) + (l – 2d) + l. + (l – 2d) + (l – 2d) + l. + (l – 2d) + (l – 2d) + l. + (l – 2d) + (l – 2d) + l. (1) Sn = l + (l – 2d) +. + (a + 2d) + a = a + a + a + a + a + a = l+ (l – 2d) + a = a + a + a = a + a + a = a + a + a = a + a = a + a = a + a = a + a = a + … (2) Equations (1) and (2) are combined to form the second equation.
plus (a+ l) plus((a+ l) plus (a+ l) plus (a + l) + (a + l) +.
2 As an example, consider the following equation: Sn=a+ln(a).
(3) In this case, the formula to get the sum of a series is S n= (n/2)(a + l), where an is the first term in the series, l is the last term in the series, and n is the number of terms in the series.
d In the case of S n, (n/2)(a + a + (n – 1)d) is the value of S n. In the case of S n= (n/2)(2a + (n – 1) x d), the formula is Observation: The successive terms in an Arithmetic Progression can alternatively be written as a3d, a2d, ad, a, a+d, a+2d, a+3d, and so on.
Finite Sequence: Definition & Examples – Video & Lesson Transcript
When there is a finite series, the first term is followed by a second term, and so on until the last term. In a finite sequence, the letternoften reflects the total number of phrases in the sequence. In a finite series, the first term may be represented by a (1), the second term can be represented by a (2), etc. Parentheses are often used to separate numbers adjacent to thea, but parentheses will be used at other points in this course to distinguish them from subscripts. This terminology is illustrated in the following graphic.
Examples
Despite the fact that there are many other forms of finite sequences, we shall confine ourselves to the field of mathematics for the time being. The prime numbers smaller than 40, for example, are an example of a finite sequence, as seen in the table below: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, Another example is the range of natural numbers between zero and one hundred. Due to the fact that it would be tedious to write down all of the terms in this finite sequence, we will demonstrate it as follows: 1, 2, 3, 4, 5,., 100 are the numbers from 1 to 100.
Alternatively, there might be alternative sequences that begin in the same way and end in 100, but which do not contain the natural numbers less than or equal to 100.
Finding Patterns
Let’s take a look at some finite sequences to see if there are any patterns.
Example 1
An anarithmetic finite sequence is a sequence in which all pairs of succeeding terms share a common difference and is defined as follows: Figure out which of these arithmetic finite sequences has the most common difference: 2, 7, 12, 17,., 47 are the prime numbers. Because the common difference is 5, the first four terms of the series demonstrate that the common difference is 5. With another way of saying it, we may add 5 to any phrase in the series to get the following term in the sequence.
Finding Sums of Finite Arithmetic Series – Sequences and Series (Algebra 2)
The sum of all terms for a finitearithmetic sequencegiven bywherea1 is the first term,dis the common difference, andnis the number of terms may be determined using the following formula: wherea1 is the first term,dis the common difference, andnis the number of terms Consider the arithmetic sum as an example of how the total is computed in this manner. If you choose not to add all of the words at once, keep in mind that the first and last terms may be recast as2fives. This method may be used to rewrite the second and secondtolast terms as well.
There are 9fives in all, and the aggregate is 9 x 5 = 9.
expand more Due to the fact that the difference between each term is constant, this sequence from 1 through 1000 is arithmetic.
The first phrase, a1, is one and the last term, is one thousand thousand. The total number of terms is less than 1000. The sum of all positive integers up to and including 1000 is 500 500. }}}!}}}Test}}} arrow back} arrow forwardarrow leftarrow right
Arithmetic Sequence – Formula, Meaning, Examples
When you have a succession of integers where the differences between every two subsequent numbers are the same, you have an arithmetic sequence. Let us take a moment to review what a sequence is. A sequence is a set of integers that are arranged in a certain manner. An arithmetic sequence is defined as follows: 1, 6, 11, 16,. is an arithmetic sequence because it follows a pattern in which each number is acquired by adding 5 to the phrase before it. There are two arithmetic sequence formulae available.
 The formula for determining the nth term of an arithmetic series. An arithmetic series has n terms, and the sum of the first n terms is determined by the following formula:
Let’s look at the definition of an arithmetic sequence, as well as arithmetic sequence formulae, derivations, and a slew of other examples to get us started.
1.  What is an Arithmetic Sequence? 
2.  Terms Related to Arithmetic Sequence 
3.  Nth Term of Arithmetic Sequence Formula 
4.  Sum of Arithmetic sequence Formula 
5.  Arithmetic Sequence Formulas 
6.  Difference Between Arithmetic and Geometric Sequence 
7.  FAQs on Arithmetic sequence 
What is an Arithmetic Sequence?
There are two ways in which anarithmetic sequence can be defined. When the differences between every two succeeding words are the same, it is said to be in sequence (or) Every term in an arithmetic series is generated by adding a specified integer (either positive or negative, or zero) to the term before it. Here is an example of an arithmetic sequence.
Arithmetic Sequences Example
For example, consider the series 3, 6, 9, 12, 15, which is an arithmetic sequence since every term is created by adding a constant number (3) to the term immediately before that one. Here,
 A = 3 for the first term
 D = 6 – 3 for the common difference
 12 – 9 for the second term
 15 – 12 for the third term
 A = 3 for the third term
As a result, arithmetic sequences can be expressed as a, a + d, a + 2d, a + 3d, and so forth. Let’s use the previous scenario as an example of how to test this pattern. a, a + d, a + 2d, a + 3d, a + 4d,. = 3, 3 + 3, 3 + 2(3), 3 + 3(3), 3 + 4(3),. = 3, 6, 9, 12, 15,. = 3, 6, 9, 12, 15,. = 3, 6, 9, 12, 15,. = 3, 6, 9, 12, 15,. Here are a few more instances of arithmetic sequences to consider:
 5, 8, 11, 14,
 80, 75, 70, 65, 60,
 2/2, 3/2, 2/2,
 2, 22, 32, 42,
 5/8, 11/14,
The terms of an arithmetic sequence are often symbolized by the letters a1, a2, a3, and so on. Arithmetic sequences are discussed in the following way, according to the vocabulary we employ.
First Term of Arithmetic Sequence
The first term of an arithmetic sequence is, as the name implies, the first integer in the sequence. It is often symbolized by the letters a1 (or) a. For example, the first word in the sequence 5, 8, 11, 14, is the number 5. Specifically, a1 = 6 (or) a = 6.
Common Difference of Arithmetic Sequence
The addition of a fixed number to each preceding term in an arithmetic series, with one exception (the first term), has previously been demonstrated in prior sections. The “fixed number” in this case is referred to as the “common difference,” and it is symbolized by the letter d. The formula for the common difference isd = a – an1.
Nth Term of Arithmetic Sequence Formula
The addition of a fixed number to each preceding term in an arithmetic series, with one exception (the first term), has already been established. ‘d’ is used to signify the “fixed number,” which is also known as the “common difference.” The formula for the common difference is d = a – 1 and the common difference is denoted by ‘d.
Arithmetic sequence  First Term(a)  Common Difference(d)  n thtermaₙ = a₁ + (n – 1) d 

80, 75, 70, 65, 60,.  80  5  80 + (n – 1) (5)= 5n + 85 
π/2, π, 3π/2, 2π,.  π/2  π/2  π/2 + (n – 1) (π/2)= nπ/2 
√2, 2√2, 3√2, 4√2,.  √2  √2  √2 + (n – 1) (√2)= √2 n 
Arithmetic Sequence Recursive Formula
It is possible to utilize the following formula for finding the nthterm of an arithmetic series in order to discover any term of that sequence if the values of ‘a1′ and’d’ are known, however this is not recommended. One further method of determining what term is the n thterm is to utilize the ” recursive formula of an arithmetic sequence “. This formula may be used to determine the next term (an) of an arithmetic sequence given both its preceding term (an1) and the value of the variable ‘d’ are known.
Example: If a19 = 72 and d = 7, find the value of a21 in an arithmetic sequence. Solution: a20 = a19 + d = 72 + 7 = 65 is obtained by applying the recursive formula. a21 = a20 + d = 65 + 7 = 58; a21 = a20 + d = 65 + 7 = 58; a21 = a20 + d = 65 + 7 = 58; As a result, the value of a21 is 58.
Sum of Arithmetic sequence Formula
To obtain the sum of the first n terms of an arithmetic sequence, the sum of the arithmetic sequence formula is employed. Consider the following arithmetic sequence: a1 (or ‘a’) is the first term, and d is the common difference between the first and second terms. Sn is the symbol for the sum of the first n terms in the expression. Then
 The following is true: When the n thterm is unknown, Sn= n/2
 When the n thterm is known, Sn= n/2
Example Ms. Natalie makes $200,000 each year, with an annual pay rise of $25,000 in addition to that. So, how much money does she have at the conclusion of the first five years of her career? Solution In Ms. Natalie’s first year of employment, she earns a sum equal to a = 2,000,000. The annual increase is denoted by the symbol d = 25,000. We need to figure out how much money she will make in the first five years. As a result, n = 5. In the sum sum of arithmetic sequence formula, substituting these numbers results in Sn = n/2 Sn = 5/2(2(200000) + (5 – 1)(25000), which is 5/2 (400000 +100000), which is equal to 5/2 (500000), which is equal to 1250000.
We may modify this formula to be more useful for greater values of the constant ‘n.’
Sum of Arithmetic Sequence Proof
Consider the following arithmetic sequence: a1 is the first term, and d is the common difference between the two terms. The sum of the first ‘n’ terms of the series is given bySn = a1 + (a1 + d) + (a1 + 2d) +. + an, where Sn = a1 + (a1 + d) + (a1 + 2d) +. + an. (1) Let us write the same total from right to left in the same manner (i.e., from the n thterm to the first term). (an – d) + (an – 2d) +. + a1. Sn = a plus (an – d) plus (an – 2d) +. + a1. (2)By combining (1) and (2), all words beginning with the letter ‘d’ are eliminated.
+ (a1 + an) 2Sn = n (a1 + an) = n (a1 + an) Sn =/2 is a mathematical expression.
Arithmetic Sequence Formulas
The following are the formulae that are connected to the arithmetic sequence.
 There is a common distinction, the nth phrase, a = (a + 1)d
 The sum of n terms, Sn =/2 (or) n/2 (2a + 1)d
 The nth term, a = (a + 1)d
 The nth term, a = a + (n1)d
Difference Between Arithmetic and Geometric Sequence
The following are the distinctions between arithmetic sequence and geometric sequence:
Arithmetic sequences  Geometric sequences 
In this, the differences between every two consecutive numbers are the same.  In this, theratiosof every two consecutive numbers are the same. 
It is identified by the first term (a) and the common difference (d).  It is identified by the first term (a) and the common ratio (r). 
There is a linear relationship between the terms.  There is an exponential relationship between the terms. 
Notes on the Arithmetic Sequence that are very important
 Arithmetic sequences have the same difference between every two subsequent numbers
 This is known as the difference between two consecutive numbers. The common difference of an arithmetic sequence a1, a2, and a3 is d = a2 – a1 = a3 – a2 =
 The common difference of an arithmetic sequence a1, a2, and a3 is d = a2 – a1 = a3 – a2 =
 It is an= a1 + (n1)d for the nth term of an integer arithmetic sequence. It is equal to n/2 when the sum of the first n terms of an arithmetic sequence is calculated. Positive, negative, or zero can be used to represent the common difference of arithmetic sequences.
Arithmetic SequenceRelated Discussion Topics
 Calculator of Sequences
 Calculator for series
 Calculator for the Arithmetic Sequence
 Calculator for Geometric Sequences
Solved Examples on Arithmetic Sequence
 Examples: Find the nth term in the arithmetic sequence 5, 7/2, 2 and the nth term in the arithmetic sequence Solution: The numbers in the above sequence are 5, 7/2, 2, and. There are two terms in this equation: the first is equal to 5, and the common difference is equal to (7/2) – (5) = 2 – (7/2) = 3/2. The n thterm of an arithmetic sequence can be calculated using the formulaan = a + b. (n – 1) dan = 5 +(n – 1) (3/2)= 5+ (3/2)n – 3/2= 3n/2 – 13/2 = dan = 5 +(n – 1) (3/2)= dan = 5 +(n – 1) (3/2)= dan = 5 +(n – 1) (3/2)= dan = 5 +(3/2)n – 3/2= dan = 5 +(3/2)n – 3/2= dan = Example 2:Which term of the arithmetic sequence 3, 8, 13, 18, the answer is: the specified arithmetic sequence is: 3, 8, 13, 18, and so on. The first term is represented by the symbol a = 3. The common difference is d = 8 – (3) = 13 – (8) = 5. The common difference is d = 8 – (3) = 13 – (8) = 5. It has been established that the n thterm is a = 248. All of these values should be substituted in the n th l term of an arithmetic sequence formula,an = a + b. (n – 1) d248 equals 3 plus (5) (n – 1) the sum of 248 and 248 equals 3 5n, and the sum of 5n and 250 equals 5nn equals 50. Answer: The number 248 represents the 50th phrase in the provided sequence.
Continue to the next slide proceed to the next slide Simple graphics might help you break through difficult topics. Math will no longer be a difficult topic for you, especially if you visualize the concepts and grasp them as a result. Schedule a NoObligation Trial Class.
FAQs on Arithmetic sequence
An arithmetic sequence is a sequence of integers in which every term (with the exception of the first term) is generated by adding a constant number to the preceding term. For example, the arithmetic sequence 1, 3, 5, 7, is an arithmetic sequence because each term is created by adding 2 (a constant integer) to the term before it.
What are Arithmetic Sequence Formulas?
Here are the formulae connected to an arithmetic series where a1 (or a) is the first term and d is the common difference: a1 (or a) is the first term, and d is the common difference:
 When we look at the common difference, it is second term minus first term. The n thterm of the series is defined as a = a + (n – 1)d
 Sn =/2 (or) n/2 (2a + (n – 1)d) is the sum of the n terms in the sequence.
How to Find An Arithmetic Sequence?
Whenever the difference between every two successive terms of a series is the same, then the sequence is said to be an arithmetic sequence. For example, the numbers 3, 8, 13, and 18 are arithmetic because
What is the n thterm of an Arithmetic Sequence Formula?
The n thterm of arithmetic sequences is represented by the expression a = a + (n – 1) d. The letter ‘a’ stands for the first term, while the letter ‘d’ stands for the common difference.
What is the Sum of an Arithmetic Sequence Formula?
Arithmetic sequences with a common difference ‘d’ and the first term ‘a’ are denoted by Sn, and we have two formulae to compute the sum of the first n terms with the common difference ‘d’.
What is the Formula to Find the Common Difference of Arithmetic sequence?
As the name implies, the common difference of an arithmetic sequence is the difference between every two of its consecutive (or consecutively occurring) terms. Finding the common difference of an arithmetic series may be calculated using the formula: d = a – an1.
How to Find n in Arithmetic sequence?
When we are asked to find the number of terms (n) in arithmetic sequences, it is possible that part of the information about a, d, an, or Sn has already been provided in the problem. We will simply substitute the supplied values in the formulae of an or Sn and solve for n as a result of this.
How To Find the First Term in Arithmetic sequence?
The number that appears in the first position from the left of an arithmetic sequence is referred to as the first term of the sequence. It is symbolized by the letter ‘a’. If the letter ‘a’ is not provided in the problem, then the problem may contain some information concerning the letter d (or) the letter a (or) the letter Sn. We shall simply insert the specified values in the formulae of an or Sn and solve for a by dividing by two.
What is the Difference Between Arithmetic Sequence and Arithmetic Series?
When it comes to numbers, an arithmetic sequence is a collection in which all of the differences between every two successive integers are equal to one, and an arithmetic series is the sum of a few or more terms of an arithmetic sequence.
What are the Types of Sequences?
In mathematics, there are three basic types of sequences. They are as follows:
 The succession of arithmetic operations
 Sequence of geometric figures
 Sequence of harmonic notes
What are the Applications of Arithmetic Sequence?
Here are some examples of applications: The pay of a person who receives an annual raise of a fixed amount, the rent of a taxi that charges by the mile traveled, the number of fish in a pond that increases by a certain number each month, and so on are examples of steady increases.
How to Find the n thTerm in Arithmetic Sequence?
The following are the actions to take in order to get the n thterm of arithmetic sequences:
 Identify the first term in the sentence, a
 D is an abbreviation for common difference. Choose the word that you wish to use. n, to be precise. All of them should be substituted into the formula a = a + (n – 1) d
How to Find the Sum of n Terms of Arithmetic Sequence?
To get the sum of the first n terms of arithmetic sequences, use the following formula:
 Identify the initial term (a)
 The common difference (d)
 And the last term (e). Determine which phrase you wish to use (n)
 All of them should be substituted into the formula Sn= n/2(2a + (n – 1)d)
Arithmetic Sequence: Formula & Definition – Video & Lesson Transcript
Afterwards, the th term in a series will be denoted by the symbol (n). The first term of a series is a (1), and the 23rd term of a sequence is the letter a (1). (23). Parentheses will be used at several points in this course to indicate that the numbers next to thea are generally written as subscripts.
Finding the Terms
Let’s start with a straightforward problem. We have the following numbers in our sequence: 3, 2, 7, 12,. What is the seventh and last phrase in this sequence? As we can see, the most typical difference between successive periods is five points.
The fourth term equals twelve, therefore a (4) = twelve. We can continue to add terms to the list in the following order until we reach the seventh term: 3, 2, 7, 12, 17, 22, 27,. and so on. This tells us that a (7) = 27 is the answer.
Finding then th Term
Consider the identical sequence as in the preceding example, with the exception that we must now discover the 33rd word oracle (33). We may utilize the same strategy as previously, but it would take a long time to complete the project. We need to come up with a way that is both faster and more efficient. We are aware that we are starting with ata (1), which is a negative number. We multiply each phrase by 5 to get the next term. To go from a (1) to a (33), we’d have to add 32 consecutive terms (33 – 1 = 32) to the beginning of the sequence.
To put it another way, a (33) = 3 + (33 – 1)5.
a (33) = 3 + (33 – 1)5 = 3 + 160 = 157.
Then the relationship between the th term and the initial terma (1) and the common differencedis provided by:
Arithmetic progression – Wikipedia
The evolution of mathematical operations The phrase “orarithmetic sequence” refers to a sequence of integers in which the difference between successive words remains constant. Consider the following example: the sequence 5, 7, 9, 11, 13, 15,. is an arithmetic progression with a common difference of two. As an example, if the first term of an arithmetic progression is and the common difference between succeeding members is, then in general the th term of the series () is given by:, and in particular, A finite component of an arithmetic progression is referred to as a finite arithmetic progression, and it is also referred to as an arithmetic progression in some cases.
Sum
2  +  5  +  8  +  11  +  14  =  40 
14  +  11  +  8  +  5  +  2  =  40 


16  +  16  +  16  +  16  +  16  =  80 
Calculation of the total amount 2 + 5 + 8 + 11 + 14 = 2 + 5 + 8 + 11 + 14 When the sequence is reversed and added to itself term by term, the resultant sequence has a single repeating value equal to the sum of the first and last numbers (2 + 14 = 16), which is the sum of the first and final numbers in the series. As a result, 16 + 5 = 80 is double the total. When all the elements of a finite arithmetic progression are added together, the result is known as anarithmetic series. Consider the following sum, for example: To rapidly calculate this total, begin by multiplying the number of words being added (in this case 5), multiplying by the sum of the first and last numbers in the progression (in this case 2 + 14 = 16), then dividing the result by two: In the example above, this results in the following equation: This formula is applicable to any real numbers and.
Derivation
An animated demonstration of the formula that yields the sum of the first two numbers, 1+2+.+n. Start by stating the arithmetic series in two alternative ways, as shown above, in order to obtain the formula. When both sides of the two equations are added together, all expressions involvingdcancel are eliminated: The following is a frequent version of the equation where both sides are divided by two: After reinserting the replacement, the following variant form is produced: Additionally, the mean value of the series may be computed using the following formula: The formula is extremely close to the mean of an adiscrete uniform distribution in terms of its mathematical structure.
Product
When the members of a finite arithmetic progression with a beginning elementa1, common differencesd, andnelements in total are multiplied together, the product is specified by a closed equation where indicates the Gamma function. When the value is negative or 0, the formula is invalid. This is a generalization of the fact that the product of the progressionis provided by the factorialand that the productforpositive integersandis supplied by the factorial.
Derivation
Where represents the factorial ascension. According to the recurrence formula, which is applicable for complex numbers0 “In order to have a positive complex number and an integer that is greater than or equal to 1, we need to divide by two. As a result, if0 “as well as a concluding note
Examples
Exemple No. 1 If we look at an example, up to the 50th term of the arithmetic progression is equal to the product of all the terms. The product of the first ten odd numbers is provided by the number = 654,729,075 in Example 2.
Standard deviation
In any mathematical progression, the standard deviation may be determined aswhere is the number of terms in the progression and is the common difference between terms. The standard deviation of an adiscrete uniform distribution is quite close to the standard deviation of this formula.
Intersections
In any mathematical progression, the standard deviation may be determined aswhere is the number of terms in the progression and is the average difference between terms. The standard deviation of an adiscrete uniform distribution is extremely similar to the expression.
History
This method was invented by a young Carl Friedrich Gaussin primary school student who, according to a story of uncertain reliability, multiplied n/2 pairs of numbers in the sum of the integers from 1 through 100 by the values of each pairn+ 1. This method is used to compute the sum of the integers from 1 through 100. However, regardless of whether or not this narrative is true, Gauss was not the first to discover this formula, and some believe that its origins may be traced back to the Pythagoreans in the 5th century BC.
See also
 This method was invented by a young Carl Friedrich Gaussin primary school student who, according to a story of uncertain reliability, multiplied n/2 pairs of numbers in the sum of the integers from 1 through 100 by the values of each pairn+ 1. The method is used to compute the sum of the integers from 1 through 100. Regardless of whether or not this narrative is true, Gauss was not the first to discover this formula, and some believe that its origins may be traced back to the Pythagoreans in the 5th century BCE. Similar rules were known in antiquity to Archimedes, Hypsicles, and Diophantus
 In China to Zhang Qiujian
 In India to Aryabhata, Brahmagupta, and Bhakara II
 And in medieval Europe to Alcuin, Dicuil, Fibonacci, and Sacrobosco, as well as to anonymous commentators on the Talmud known as Tosafists
 And in the United States to Benjamin Franklin.
References
 Duchet, Pierre (1995), “Hypergraphs,” in Graham, R. L., Grötschel, M., and Lovász, L. (eds. ), Handbook of combinatorics, Vol. 1, 2, Amsterdam: Elsevier, pp. 381–432, MR1373663. Duchet, Pierre (1995), “Hypergraphs,” in Graham, R. L., Grötschel, M., and Particularly noteworthy are Section 2.5, “Helly Property,” pages 393–394
 And Hayes, Brian (2006). “Gauss’s Day of Reckoning,” as the saying goes. Journal of the American Scientist, 94(3), 200, doi:10.1511/2006.59.200 The original version of this article was published on January 12, 2012. retrieved on October 16, 2020
 Retrieved on October 16, 2020
 “The Unknown Heritage”: a trace of a longforgotten center of mathematical expertise,” J. Hyrup, et al. The American Journal of Physics 62, 613–654 (2008)
 Tropfke, Johannes, et al (1924). Geometrie analytisch (analytical geometry) pp. 3–15. ISBN 9783111080628
 Tropfke, Johannes. Walter de Gruyter. pp. 3–15. ISBN 9783111080628
 (1979). Arithmetik and Algebra are two of the most important subjects in mathematics. pp. 344–354, ISBN 9783110048933
 Problems to Sharpen the Young,’ Walter de Gruyter, pp. 344–354, ISBN 9783110048933
 The Mathematical Gazette, volume 76, number 475 (March 1992), pages 102–126
 Ross, H.E.Knott, B.I. (2019) Dicuil (9th century) on triangle and square numbers, British Journal for the History of Mathematics, volume 34, number 2, pages 79–94
 Laurence E. Sigler is the translator for this work (2002). The Liber Abaci of Fibonacci. SpringerVerlag, Berlin, Germany, pp.259–260, ISBN 0387954198
 Victor J. Katz is the editor of this work (2016). The Mathematics of Medieval Europe and North Africa: A Sourcebook is a reference work on medieval mathematics. 74.23 A Mediaeval Derivation of the Sum of an Arithmetic Progression. Princeton, NJ: Princeton University Press, 1990, pp. 91, 257. ISBN 9780691156859
 Stern, M. (1990). 74.23 A Mediaeval Derivation of the Sum of an Arithmetic Progression. Princeton, NJ: Princeton University Press, 1990, pp. 91, 257. ISBN 9780691156859
 Stern, M. Journal of the American Mathematical Society, vol. 74, no. 468, pp. 157159. doi:10.2307/3619368.
External links
 Weisstein, Eric W., “Arithmetic series,” in Encyclopedia of Mathematics, EMS Press, 2001
 “Arithmetic progression,” in Encyclopedia of Mathematics, EMS Press, 2001. MathWorld
 Weisstein, Eric W. “Arithmetic series.” MathWorld
 Weisstein, Eric W. “Arithmetic series.”
Arithmetic Sequences
In mathematics, an arithmetic sequence is a succession of integers in which the value of each number grows or decreases by a fixed amount each term. When an arithmetic sequence has n terms, we may construct a formula for each term in the form fn+c, where d is the common difference. Once you’ve determined the common difference, you can calculate the value ofcby substituting 1fornand the first term in the series fora1 into the equation. Example 1: The arithmetic sequence 1,5,9,13,17,21,25 is an arithmetic series with a common difference of four.
 For the thenthterm, we substituten=1,a1=1andd=4inan=dn+cto findc, which is the formula for thenthterm.
 As an example, the arithmetic sequence 129630360 is an arithmetic series with a common difference of three.
 It is important to note that, because the series is decreasing, the common difference is a negative number.) To determine the next3 terms, we just keep subtracting3: 6 3=9 9 3=12 12 3=15 6 3=9 9 3=12 12 3=15 As a result, the next three terms are 9, 12, and 15.
 As a result, the formula for the fifteenth term in this series isan=3n+15.
Exemple No. 3: The number series 2,3,5,8,12,17,23,. is not an arithmetic sequence. Differencea2 is 1, but the following differencea3 is 2, and the differencea4 is 3. There is no way to write a formula in the form of forman=dn+c for this sequence. Geometric sequences are another type of sequence.
What is an arithmetic sequence? + Example
An arithmetic sequence is a series (list of numbers) in which there is a common difference (a positive or negative constant) between the items that are consecutively listed. For example, consider the following instances of arithmetic sequences: 1.) The numbers 7, 14, 21, and 28 are used because the common difference is 7. 2.) The numbers 48, 45, 42, and 39 are chosen because they have a common difference of – 3. The following are instances of arithmetic sequences that are not to be confused with them: It is not 2,4,8,16 since the difference between the first and second terms is 2, but the difference between the second and third terms is 4, and the difference between the third and fourth terms is 8 because the difference between the first and second terms is 2.
2.) The numbers 1, 4, 9, and 16 are incorrect because the difference between the first and second is 3, the difference between the second and third is 5, and the difference between the third and fourth is 7.
The reasons for this are that the difference between the first and second is three points, the difference between the second and third is two points, and the difference between third and fourth is five points.
13.2: Arithmetic Sequences
Example (PageIndex): After writing the first Term, write the second Term. An Arithmetic Sequence with a Clearly Defined Formula Create an explicit formula for the arithmetic series using the following syntax: ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( It is possible to calculate the common difference by subtracting the first term from the second term. The most noticeable change is referred to as (10). To simplify the formula, substitute the common difference and the first term in the series into it.
Drawing (figure) (PageIndex ) Take part in an exercise program (PageIndex ) For the arithmetic series that follows, provide an explicit formula for it.
Finding the Number of Terms in a Finite Arithmetic Sequence
When determining the number of terms in a finite arithmetic sequence, explicit formulas can be employed to make the determination. Finding the common difference and determining the number of times the common difference must be added to the first term in order to produce the last term of the sequence are both necessary steps. Calculate the total number of terms in a finite arithmetic sequence using the first three terms and the last term as inputs.
 Figure out what the common difference (d) is
 Replace the common difference and the first term in (a n=a 1+d(n–1)) with the common difference and the first term. Make a substitution for the final word in (a n) and solve for (n)
 A.
Figure 1: Finding the Number of Terms in a Finite Arithmetic Sequence using the PageIndex method. The number of terms in the finite arithmetic sequence has to be determined. ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( It is possible to calculate the common difference by subtracting the first term from the second term. (1−8=−7) The most often encountered difference is (7). Substitute the common difference and the first term of the sequence into the nth term formula, and then simplify the resultant formula.
There are a total of eight terms in the series.
answError: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: There are a total of eleven terms in the series.
Solving Application Problems with Arithmetic Sequences
In many application situations, it is typically preferable to utilize an initial term of (a 0) rather than (a 1) as the first term. In order to account for the variation in beginning terms in both cases, we make a little modification to the explicit formula. The following is the formula that we use: The following is an example of (PageIndex ): ProblemSolving using Arithmetic Sequences in Practical Situations Every week, a fiveyearold child is given a monetary allowance of one dollar. His parents offer him a yearly raise of ($2 per week) on top of his current salary.
 Create a method for calculating the child’s weekly stipend over the course of a year
 What will the child’s allowance be when he reaches the age of (16) years?
 In this case, an arithmetic sequence with an initial term of (1,1) and a common difference of (1,1) may be used to simulate the scenario (2). Let (A) be the amount of the allowance and n denote the number of years after the age of retirement (5). This is what we get if we use the changed explicit formula for an arithmetic sequence: (A r=1+2n)
 By subtracting, we may get the number of years that have passed since the age of (5). (16−5=11) It is our intention to obtain the child’s allowance after eleven years. In order to calculate the child’s allowance at age, substitute (11) into the calculation (16). (A_ =1+2(11)=23) is a prime number. The child’s allowance will be ($23) per week when he or she reaches the age of sixteen.
Take part in an exercise program (PageIndex ) The next week, a lady chooses to go for a tenminute run every day, with the goal of increasing the length of her daily exercise by four minutes each week. Formulate the time she will run after (n) weeks to determine the distance she will cover. How long will her daily run last in a year and a half from now? Answer The formula is (T n=10+4n,) and it will take her a total of (42) minutes to complete. In addition to further teaching and practice with arithmetic sequences, you can access this online resource for that purpose.
recursive formula for nth term of an arithmetic sequence  (a_n=a_ +d) (n≥2) 
explicit formula for nth term of an arithmetic sequence  (a_n=a_1+d(n−1)) 
 When there is a constant difference between any two consecutive terms in an arithmetic sequence, the sequence is called an arithmetic sequence
 The constant difference between two consecutive terms is known as the common difference
 The common difference is the number that is added to any one term of an arithmetic sequence in order to generate the subsequent term. See the following example: (PageIndex)
 The terms of an arithmetic sequence can be obtained by starting with the first term and adding the common difference over and over until the sequence is complete. See Examples ((PageIndex ) and ((PageIndex ) for more information. The recursive formula for an arithmetic series with common difference dd is provided by (a n=a_ +d), and (n2 is the number of steps in the sequence). See the following example: (PageIndex)
 As with any recursive formula, the first term in the series must be specified
 Otherwise, the formula will fail. It is possible to express an explicit formula for an arithmetic series with a common difference d using the formula (a n=a 1+d(n)1). See the following example: (PageIndex)
 When determining the number of words in a sequence, it is possible to apply an explicit formula. Observe the following example: (PageIndex)
 In application situations, we may slightly modify the explicit formula to (a n=a 0+dn). See the following example: (PageIndex)
Arithmetic Sequences and Series
The succession of arithmetic operations There is a series of integers in which each subsequent number is equal to the sum of the preceding number and specified constants. orarithmetic progression is a type of progression in which numbers are added together. This term is used to describe a series of integers in which each subsequent number is the sum of the preceding number and a certain number of constants (e.g., 1). an=an−1+d Sequence of Arithmetic Operations Furthermore, becauseanan1=d, the constant is referred to as the common difference.
For example, the series of positive odd integers is an arithmetic sequence, consisting of the numbers 1, 3, 5, 7, 9, and so on.
This word may be constructed using the generic terman=an1+2where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, To formulate the following equation in general terms, given the initial terma1of an arithmetic series and its common differenced, we may write: a2=a1+da3=a2+d=(a1+d)+d=a1+2da4=a3+d=(a1+2d)+d=a1+3da5=a4+d=(a1+3d)+d=a1+4d⋮ As a result, we can see that each arithmetic sequence may be expressed as follows in terms of its initial element, common difference, and index: an=a1+(n−1)d Sequence of Arithmetic Operations In fact, every generic word that is linear defines an arithmetic sequence in its simplest definition.
Example 1
Identify the general term of the above arithmetic sequence and use that equation to determine the series’s 100th term. For example: 7,10,13,16,19,… Solution: The first step is to determine the common difference, which is d=10 7=3. It is important to note that the difference between any two consecutive phrases is three. The series is, in fact, an arithmetic progression, with a1=7 and d=3. an=a1+(n1)d=7+(n1)3=7+3n3=3n+4 and an=a1+(n1)d=7+(n1)3=7+3n3=3n+4 and an=a1+(n1)d=3 As a result, we may express the general terman=3n+4 as an equation.
Take a moment to confirm that this equation accurately reflects the sequence you’ve been given. To determine the 100th term, use the following equation: a100=3(100)+4=304 Answer_an=3n+4;a100=304 It is possible that the common difference of an arithmetic series be negative.
Example 2
Identify the general term of the given arithmetic sequence and use it to determine the 75th term of the series: 6,4,2,0,−2,… Solution: Make a start by determining the common difference, d = 4 6=2. Next, determine the formula for the general term, wherea1=6andd=2 are the variables. an=a1+(n−1)d=6+(n−1)⋅(−2)=6−2n+2=8−2n As a result, an=8nand the 75thterm may be determined as follows: an=8nand the 75thterm a75=8−2(75)=8−150=−142 Answer_an=8−2n;a100=−142 The terms in an arithmetic sequence that occur between two provided terms are referred to as arithmetic means.
Example 3
Find all of the words that fall between a1=8 and a7=10. in the context of an arithmetic series Or, to put it another way, locate all of the arithmetic means between the 1st and 7th terms. Solution: Begin by identifying the points of commonality. In this situation, we are provided with the first and seventh terms, respectively: an=a1+(n−1) d Make use of n=7.a7=a1+(71)da7=a1+6da7=a1+6d Substitutea1=−8anda7=10 into the preceding equation, and then solve for the common differenced result. 10=−8+6d18=6d3=d Following that, utilize the first terma1=8.
a1=3(1)−11=3−11=−8a2=3 (2)−11=6−11=−5a3=3 (3)−11=9−11=−2a4=3 (4)−11=12−11=1a5=3 (5)−11=15−11=4a6=3 (6)−11=18−11=7} In arithmetic, a7=3(7)11=21=10 means a7=3(7)11=10 Answer: 5, 2, 1, 4, 7, and 8.
Example 4
Find the general term of an arithmetic series with a3=1 and a10=48 as the first and last terms. Solution: We’ll need a1 and d in order to come up with a formula for the general term. Using the information provided, it is possible to construct a linear system using these variables as variables. andan=a1+(n−1) d:{a3=a1+(3−1)da10=a1+(10−1)d⇒ {−1=a1+2d48=a1+9d Make use of a3=1. Make use of a10=48. Multiplying the first equation by one and adding the result to the second equation will eliminate a1.
an=a1+(n−1)d=−15+(n−1)⋅7=−15+7n−7=−22+7n Answer_an=7n−22 Take a look at this!
For example: 32,2,52,3,72,… Answer_an=12n+1;a100=51
Arithmetic Series
Series of mathematical operations When an arithmetic sequence is added together, the result is called the sum of its terms (or the sum of its terms and numbers). Consider the following sequence: S5=n=15(2n1)=++++= 1+3+5+7+9+25=25, where S5=n=15(2n1)=++++ = 1+3+5+7+9=25, where S5=n=15(2n1)=++++= 1+3+5+7+9 = 25. Adding 5 positive odd numbers together, like we have done previously, is manageable and straightforward. Consider, on the other hand, adding the first 100 positive odd numbers. This would be quite timeconsuming.
When we write this series in reverse, we get Sn=an+(and)+(an2d)+.+a1 as a result.
2.:Sn=n(a1+an) 2 Calculate the sum of the first 100 terms of the sequence defined byan=2n1 by using this formula. There are two variables, a1 and a100. The sum of the two variables, S100, is 100 (1 + 100)2 = 100(1 + 199)2.
Example 5
The sum of the first 50 terms of the following sequence: 4, 9, 14, 19, 24,. is to be found. The solution is to determine whether or not there is a common difference between the concepts that have been provided. d=9−4=5 It is important to note that the difference between any two consecutive phrases is 5. The series is, in fact, an arithmetic progression, and we may writean=a1+(n1)d=4+(n1)5=4+5n5=5n1 as an anagram of the sequence. As a result, the broad phrase isan=5n1 is used. For this sequence, we need the 1st and 50th terms to compute the 50thpartial sum of the series: a1=4a50=5(50)−1=249 Then, using the formula, find the partial sum of the given arithmetic sequence that is 50th in length.
Example 6
Evaluate:Σn=135(10−4n). This problem asks us to find the sum of the first 35 terms of an arithmetic series with a general terman=104n. The solution is as follows: This may be used to determine the 1 stand for the 35th period. a1=10−4(1)=6a35=10−4(35)=−130 Then, using the formula, find out what the 35th partial sum will be. Sn=n(a1+an)2S35=35⋅(a1+a35)2=352=35(−124)2=−2,170 2,170 is the answer.
Example 7
In an outdoor amphitheater, the first row of seating comprises 26 seats, the second row contains 28 seats, the third row contains 30 seats, and so on and so forth. Is there a maximum capacity for seating in the theater if there are 18 rows of seats? The Roman Theater (Fig. 9.2) (Wikipedia) Solution: To begin, discover a formula that may be used to calculate the number of seats in each given row. In this case, the number of seats in each row is organized into a sequence: 26,28,30,… It is important to note that the difference between any two consecutive words is 2.
where a1=26 and d=2.
As a result, the number of seats in each row may be calculated using the formulaan=2n+24.
In order to do this, we require the following 18 thterms: a1=26a18=2(18)+24=60 This may be used to calculate the 18th partial sum, which is calculated as follows: Sn=n(a1+an)2S18=18⋅(a1+a18)2=18(26+60) 2=9(86)=774 There are a total of 774 seats available.
Calculate the sum of the first 60 terms of the following sequence of numbers: 5, 0, 5, 10, 15,.
Answer:S60=−8,550
Key Takeaways
 When the difference between successive terms is constant, a series is called an arithmetic sequence. According to the following formula, the general term of an arithmetic series may be represented as the sum of its initial term, common differenced term, and indexnumber, as follows: an=a1+(n−1)d
 An arithmetic series is the sum of the terms of an arithmetic sequence
 An arithmetic sequence is the sum of the terms of an arithmetic series
 As a result, the partial sum of an arithmetic series may be computed using the first and final terms in the following manner: Sn=n(a1+an)2
Topic Exercises
 Given the first term and common difference of an arithmetic series, write the first five terms of the sequence. Find a formula that describes the generic term. The values of a1 are 5 and 3
 12 and 2
 15 and 5
 7 and 4 respectively
 12 and 1
 A1=23 and 13 respectively
 1 and 12 respectively
 A1=54 and 14. The values of a1 are 1.8 and 0.6
 4.3 and 2.1
 And a1=5.4 and 2.1 respectively.
 Locate a formula for the general term and apply it to get the 100 thterm, given the arithmetic series given the sequence 0.8, 2, 3.2, 4.4, 5.6,.
 4.4, 7.5, 13.7, 16.8,.
 3, 8, 13, 18, 23,.
 3, 7, 11, 15, 19,.
 6, 14, 22, 30, 38,.
 5, 10, 15, 20, 25,.
 2, 4, 6, 8, 10,.
 12,52,92,132,.
 13, 23, 53,83,.
 14,12,54,2,114,. Find the positive odd integer that is 50th
 Find the positive even integer that is 50th
 Find the 40 th term in the sequence that consists of every other positive odd integer in the following format: 1, 5, 9, 13,.
 Find the 40th term in the sequence that consists of every other positive even integer: 1, 5, 9, 13,.
 Find the 40th term in the sequence that consists of every other positive even integer: 2, 6, 10, 14,.
 2, 6, 10, 14,. What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
 What number is the phrase 172 in the arithmetic sequence 4, 4, 12, 20, 28,.
 What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
 Find an equation that yields the general term in terms of a1 and the common differenced given the arithmetic sequence described by the recurrence relationan=an1+5wherea1=2 andn1 and the common differenced
 Find an equation that yields the general term in terms ofa1and the common differenced, given the arithmetic sequence described by the recurrence relationan=an19wherea1=4 andn1
 This is the problem.
 Find a formula for the general term from the terms of an arithmetic sequence given the terms of the series. 1 = 6 and 7 = 42
 1 = 12 and 12= 6
 1 = 19 and 26 = 56
 1 = 9 and 31 = 141
 1 = 16 and 10 = 376
 1 = 54 and 11 = 654. 1 = 6 and 7 = 42
 1= 9 and 31 = 141
 1 = 6 and 7
 Find all of the arithmetic means that exist between the two supplied terms. a1=3anda6=17
 A1=5anda5=7
 A2=4anda8=7
 A5=12anda9=72
 A5=15anda7=21
 A6=4anda11=1
Part B: Arithmetic Series
 In light of the general term’s formula, figure out how much the suggested total is. an=3n+5
 S100
 An=5n11
 An=12n
 S70
 An=132n
 S120
 An=12n34
 S20
 An=n35
 S150
 An=455n
 S65
 An=2n48
 S95
 An=4.41.6n
 S75
 An=6.5n3.3
 S67
 An=3n+5
 Evaluate. 1160(3n)
 1121(2n)
 1250(4n3)
 1120(2n+12)
 170 (198n)
 1220(5n)
 160(5212n)
 151(38+14
 1120(1.5n+2.6)
 1175(0.2N1.6)
 1170 (19 The total of all 200 positive integers is found by counting them up. To solve this problem, find the sum of the first 400 positive integers.
 The generic term for a sequence of positive odd integers is denoted byan=2n1 and is defined as follows: Furthermore, the generic phrase for a sequence of positive even integers is denoted by the number an=2n. Look for the following. The sum of the first 50 positive odd numbers
 The sum of the first 200 positive odd integers
 The sum of the first 500 positive odd integers
 The sum of the first 50 positive even numbers
 The sum of the first 200 positive even integers
 The sum of the first 500 positive even integers
 The sum of the firstk positive odd integers
 The sum of the firstk positive odd integers the sum of the firstk positive even integers
 The sum of the firstk positive odd integers
 There are eight seats in the front row of a tiny theater, which is the standard configuration. Following that, each row contains three additional seats than the one before it. How many total seats are there in the theater if there are 12 rows of seats? In an outdoor amphitheater, the first row of seating comprises 42 seats, the second row contains 44 seats, the third row contains 46 seats, and so on and so forth. When there are 22 rows, how many people can fit in the theater’s entire seating capacity? The number of bricks in a triangle stack are as follows: 37 bricks on the bottom row, 34 bricks on the second row and so on, ending with one brick on the top row. What is the total number of bricks in the stack
 Each succeeding row of a triangle stack of bricks contains one fewer brick, until there is just one brick remaining on the top of the stack. Given a total of 210 bricks in the stack, how many rows does the stack have? A wage contract with a 10year term pays $65,000 in the first year, with a $3,200 raise for each consecutive year after. Calculate the entire salary obligation over a tenyear period (see Figure 1). In accordance with the hour, a clock tower knocks its bell a specified number of times. The clock strikes once at one o’clock, twice at two o’clock, and so on until twelve o’clock. A day’s worth of time is represented by the number of times the clock tower’s bell rings.
Part C: Discussion Board
 Is the Fibonacci sequence an arithmetic series or a geometric sequence? How to explain: Using the formula for the then th partial sum of an arithmetic sequenceSn=n(a1+an)2and the formula for the general terman=a1+(n1)dto derive a new formula for the then th partial sum of an arithmetic sequenceSn=n2, we can derive the formula for the then th partial sum of an arithmetic sequenceSn=n2. How would this formula be beneficial in the given situation? Explain with the use of an example of your own creation
 Discuss strategies for computing sums in situations when the index does not begin with one. For example, n=1535(3n+4)=1,659
 N=1535(3n+4)=1,659
 Carl Friedrich Gauss is the subject of a wellknown tale about his misbehaving in school. As a punishment, his instructor assigned him the chore of adding the first 100 integers to his list of disciplinary actions. According to folklore, young Gauss replied accurately within seconds of being asked. The question is, what is the solution, and how do you believe he was able to come up with the figure so quickly?
Answers
 An=3n+2
 An=5n+3
 An=6n
 An=3n+2
 An=6n+3
 An=6n+2
 1,565,450, 2,500,450, k2,
 90,800, k4,230,
 38640, 124,750,
 18,550, k765
 10,578
 20,100,
 2,500,550, k2,
 294 seats, 247 bricks, $794,000, and so on.