a_{n}_{+}_{1}= a_{n} + d, d — . : , d.
Contents
 1 How do you explain an arithmetic sequence?
 2 What is arithmetic sequence and example?
 3 Why is it an arithmetic sequence?
 4 What is the meaning of arithmetic means?
 5 What is the arithmetic mean between 10 and 24?
 6 How are arithmetic sequences related to real life?
 7 Arithmetic Sequences and Sums
 8 Arithmetic Sequence
 9 Advanced Topic: Summing an Arithmetic Series
 10 Footnote: Why Does the Formula Work?
 11 Arithmetic Sequence – Formula, Meaning, Examples
 12 What is an Arithmetic Sequence?
 13 Nth Term of Arithmetic Sequence Formula
 14 Sum of Arithmetic sequence Formula
 15 Arithmetic Sequence Formulas
 16 Difference Between Arithmetic and Geometric Sequence
 17 Solved Examples on Arithmetic Sequence
 18 FAQs on Arithmetic sequence
 18.1 What are Arithmetic Sequence Formulas?
 18.2 How to Find An Arithmetic Sequence?
 18.3 What is the n thterm of an Arithmetic Sequence Formula?
 18.4 What is the Sum of an Arithmetic Sequence Formula?
 18.5 What is the Formula to Find the Common Difference of Arithmetic sequence?
 18.6 How to Find n in Arithmetic sequence?
 18.7 How To Find the First Term in Arithmetic sequence?
 18.8 What is the Difference Between Arithmetic Sequence and Arithmetic Series?
 18.9 What are the Types of Sequences?
 18.10 What are the Applications of Arithmetic Sequence?
 18.11 How to Find the n thTerm in Arithmetic Sequence?
 18.12 How to Find the Sum of n Terms of Arithmetic Sequence?
 19 Arithmetic progression – Wikipedia
 20 Sum
 21 Product
 22 Standard deviation
 23 Intersections
 24 History
 25 See also
 26 References
 27 External links
 28 Arithmetic Sequences and Series
 29 Arithmetic Series
 30 Arithmetic Sequences
 31 Arithmetic Sequences and Series
 32 What is an Arithmetic Sequence?
 33 How an Arithmetic Sequence Works
 34 Arithmetic Mean
 35 Other Types of Sequences
 36 Arithmetic Sequences and Series – MathBitsNotebook(A2
How do you explain an 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 is arithmetic sequence and example?
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. If we add or subtract by the same number each time to make the sequence, it is an arithmetic sequence.
Why is it an arithmetic sequence?
What is an arithmetic sequence? For many of the examples above, the pattern involves adding or subtracting a number to each term to get the next term. Sequences with such patterns are called arithmetic sequences. In an arithmetic sequence, the difference between consecutive terms is always the same.
What is the meaning of arithmetic means?
The arithmetic mean is the simplest and most widely used measure of a mean, or average. It simply involves taking the sum of a group of numbers, then dividing that sum by the count of the numbers used in the series. For example, take the numbers 34, 44, 56, and 78.
What is the arithmetic mean between 10 and 24?
Using the average formula, get the arithmetic mean of 10 and 24. Thus, 10+24/2 =17 is the arithmetic mean.
Arithmetic sequences are used in daily life for different purposes, such as determining the number of audience members an auditorium can hold, calculating projected earnings from working for a company and building wood piles with stacks of logs.
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
 1, 4, 7, 10, 13, 16, 19, 22, and 25 are the first four digits of the number 1. The numbers in this sequence are separated by a factor of three. Each time the pattern is repeated, the last number is increased by three, as follows: The following is a general arithmetic sequence that we might write:
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:
 3, 8, 13, 18, 23, 28, 33, and 38 are the numbers 3, 8, 13, 18, 23, 28, 33, and 38 respectively. Each number in this series differs by a factor of five. There are two values associated with aandd, which are:
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.
In this equation, A = 1 (the first term), d = 3 (the “common difference” across terms), and n = 10 (the number of terms to add up).
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) 
Check out why the formula works since we’ll be employing an unusual “trick” that’s well worth your time to learn about. As a starting point, we’ll refer to the entire total as “S.” In S, the sum of the squares of the first and second roots of the first and second roots of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root Afterwards, rewrite S in the other direction: The formula for S is: (a + n1)d)+ (a + n2)d)+.
+(a + d)+.+(a + d)+. S = (a + n1)d)+. Now, term by term, combine the following two phrases.
Arithmetic Sequence – Formula, Meaning, Examples
Check out why the formula works since we’ll be employing an unusual “trick” that’s well worth understanding. First, we shall 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 + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + Afterwards, rewrite S in the opposite order: S = (a + (n1)d)+ (a + (n2)d)+ (a + (n3)d)+ (a + (n2)d)+. +(a + d)+a Now, term by term, combine the following two:
 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
In such case, the thterm of an arithmetic series of the form A1, A2, A3,. is given byan = a1 + (n1) d. This is also referred to as the broad word for the arithmetic sequence in some circles. This comes immediately from the notion that the arithmetic sequence a1, a2, a3,. = a1, a1 + d, a1 + 2d, a1 + 3d,. = a1, a1 + d, a1 + 3d,. = a1, a1 + 3d,. = a1, a1 + 3d,. Several arithmetic sequences are shown in the following table, along with the first term, the common difference, and the subsequent n thterms.
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
 To obtain the sum of the first n terms of an arithmetic sequence, the sum of the arithmetic sequence formula is utilized. 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 its first n terms. Then
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
 The Arithmetic Sequence: Important Points to Keep in Mind
Arithmetic SequenceRelated Discussion Topics
 Sequence Calculator, Series Calculator, Arithmetic Sequence Calculator, Geometric Sequence Calculator are all terms used to refer to the same thing.
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.
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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 arithmetic series, the geometric sequence, and the harmonic sequence are all examples of sequences.
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, a
 The common difference, d
 And the last term, e. 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 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.
Anarithmetic series is the sum of a finite arithmetic progression, and it is composed of the numbers 0 through 9.
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
The intersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression, which may be obtained using the Chinese remainder theorem. The intersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression. Whenever a pair of progressions in a family of doubly infinite arithmetic progressions has a nonempty intersection, there exists a number that is common to all of them; in other words, infinite arithmetic progressions form a Helly family.
However, it is possible that the intersection of infinitely many infinite arithmetic progressions is a single number rather than being an endless progression in and of itself.
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
 Geometric progression
 Harmonic progression
 Arithmetic progression
 Number with three sides
 Triangular number
 Sequence of arithmetic and geometry operations
 Inequality between the arithmetic and geometric means
 In mathematical progression, primes are used. Equation of difference in a linear form
 A generalized arithmetic progression is a set of integers that is formed in the same way that an arithmetic progression is, but with the addition of the ability to have numerous different differences
 Heronian triangles having sides that increase in size as the number of sides increases
 Mathematical problems that include arithmetic progressions
 Utonality
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)
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 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 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
Find the general term of an arithmetic series with a3=1 and a10=48 as the first and last elements. Solution: We’ll need a1 and d in order to come up with a formula for the broad term. With the information provided, it is possible to construct a linear system with these as variables. andan=a1+(n−1) d:{a3=a1+(3−1)da10=a1+(10−1)d⇒ {−1=a1+2d48=a1+9d In this case, the value a3=1 is appropriate. A10=48 should be used. Add the result of multiplying the first equation by one to the second equation to eliminate a1.
an=a1+(n−1)d=−15+(n−1)⋅7=−15+7n−7=−22+7n Answer_an=7n−22 You should try it. Identify the general term of the given arithmetic sequence and use that equation to compute the series’s 100th term. 32,2,52,3,72,… Answer_an=12n+1;a100=51
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
Evaluate:Σn=135(10−4n). Solved:In this scenario, we are requested to find the sum of the first 35 terms of an arithmetic series having a general terman of 104n. The 1 stand the 35thterm is determined by using this. a1=10−4(1)=6a35=10−4(35)=−130 Then, using the formula, find out what the 35th partial sum equals to. Sn=n(a1+an)2S35=35⋅(a1+a35)2=352=35(−124)2=−2,170 2,170 is the correct response.
Key Takeaways
 Evaluate:Σn=135(10−4n). We are requested to find the sum of the first 35 terms of an arithmetic series with a generic terman=104n in this example. This may be used to calculate the 1st stand for the 35th term. a1=10−4(1)=6a35=10−4(35)=−130 Then, using the formula, find out what the 35th partial sum is. Sn=n(a1+an)2S35=35⋅(a1+a35)2=352=35(−124)2=−2,170 2,170 is the correct answer.
Topic Exercises
 Given the first term and common difference of an arithmetic series, write the first five terms of the sequence. Calculate the general term for the following numbers: a1=5
 D=3
 A1=12
 D=2
 A1=15
 D=5
 A1=7
 D=4
 D=1
 A1=23
 D=13
 A 1=1
 D=12
 A1=54
 D=14
 A1=1.8
 D=0.6
 A1=4.3
 D=2.1
 Find a formula for the general term based on the arithmetic sequence and apply it to get the 100 th term based on the series. 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.
 Calculate a formula for the general term based on the terms of an arithmetic sequence: a1=6anda7=42
 A1=12anda12=6
 A1=19anda26=56
 A1=9anda31=141
 A1=16anda10=376
 A1=54anda11=654
 A3=6anda26=40
 A3=16andananda15=
 Find all possible arithmetic means between the given terms: a1=3anda6=17
 A1=5anda5=7
 A2=4anda8=7
 A5=12anda9=72
 A5=15anda7=21
 A6=4anda11=1
 A7=4anda11=1
Part B: Arithmetic Series
 Make a calculation for the provided total based on the formula for the general term 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
 Consider the following values: n=1160(3n)
 N=1121(2n)
 N=1250(4n3)
 N=1120(2n+12)
 N=170(198n)
 N=1220(5n)
 N=160(5212n)
 N=151(38n+14)
 N=1120(1.5n2.6)
 N=1175(0.2n1.6)
 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 integers
 The sum of the first 200 positive odd integers
 The sum of the first 50 positive even integers
 The sum of the first 200 positive even integers
 The sum of the first 100 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
 5, 8, 11, 14, 17
 An=3n+2
 15, 10, 5, 0, 0
 An=205n
 12,32,52,72,92
 An=n12
 1,12, 0,12, 1
 An=3212n
 1.8, 2.4, 3, 3.6, 4.2
 An=0.6n+1.2
 An=6n3
 A100=597
 An=14n
 A100=399
 An=5n
 A100=500
 An=2n32
 2,450, 90, 7,800, 4,230, 38,640, 124,750, 18,550, 765, 10,000, 20,100, 2,500, 2,550, K2, 294 seats, 247 bricks, $794,000, and
Arithmetic Sequences
In mathematics, an arithmetic sequence is a sequence of numbers in which the value of each number increases or decreases by a constant amount each term. When an arithmetic sequence has n terms, we can write 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.
Arithmetic Sequences and Series
An arithmetic sequence is a set of integers in which the difference between the words that follow is always the same as its predecessor.
Learning Objectives
Make a calculation for the nth term of an arithmetic sequence and then define the characteristics of arithmetic sequences.
Key Takeaways
 When the common differenced is used, the behavior of the arithmetic sequence is determined. Arithmetic sequences may be either limited or infinite in length.
Key Terms
 Arithmetic sequence: An ordered list of numbers in which the difference between the subsequent terms is constant
 Endless: An ordered list of numbers in which the difference between the consecutive terms is infinite
 Infinite, unending, without beginning or end
 Limitless
 Innumerable
For example, an arithmetic progression or arithmetic sequence is a succession of integers in which the difference between the following terms is always the same as the difference between the previous terms. A common difference of 2 may be found in the arithmetic sequence 5, 7, 9, 11, 13, cdots, which is an example of an arithmetic sequence.
 1: The initial term in the series
 D: The difference between the common differences of consecutive terms
 A 1: a n: Then the nth term in the series.
The behavior of the arithmetic sequence is determined by the common differenced arithmetic sequence. If the common difference,d, is the following:
 Positively, the sequence will continue to develop towards infinity (+infty). If the sequence is negative, it will regress towards negative infinity (infty)
 If it is positive, it will regress towards positive infinity (infty).
It should be noted that the first term in the series can be thought of asa 1+0cdot d, the second term can be thought of asa 1+1cdot d, and the third term can be thought of asa 1+2cdot d, and therefore the following equation givesa n:a n In the equation a n= a 1+(n1)cdot D Of course, one may always type down each term until one has the term desired—but if one need the 50th term, this can be timeconsuming and inefficient.
What is an Arithmetic Sequence?
Sequences of numbers are useful in algebra because they allow you to see what occurs when something keeps becoming larger or smaller over time. The common difference, which is the difference between one number and the next number in the sequence, is the defining characteristic of an arithmetic sequence. This difference is a constant value in arithmetic sequences, and it can be either positive or negative in nature. Consequently, an arithmetic sequence continues to grow or shrink by a defined amount each time a new number is added to the list of numbers that make up the sequence is added to it.
TL;DR (Too Long; Didn’t Read)
As defined by the Common Difference formula, an arithmetic sequence is a list of integers in which consecutive entries differ by the same amount, called the common difference. Whenever the common difference is positive, the sequence continues to grow by a predetermined amount, and when it is negative, the series begins to shrink. The geometric series, in which terms differ by a common factor, and the Fibonacci sequence, in which each number is the sum of the two numbers before it, are two more typical sequences that might be encountered.
How an Arithmetic Sequence Works
There are three elements that form an arithmetic series: a starting number, a common difference, and the number of words in the sequence. For example, the first twelve terms of an arithmetic series with a common difference of three and five terms are 12, 15, 18, 21, and 24. A declining series starting with the number 3 has a common difference of 2 and six phrases, and it is an example of a decreasing sequence. This series is composed of the numbers 3, 1, 1, 3, 5, and 7.
There is also the possibility of an unlimited number of terms in arithmetic sequences. Examples of infinite number of terms include, for example, the first sequence above with 12 terms, followed by 15 terms, followed by 18 terms, and so on infinity.
Arithmetic Mean
A matching series to an arithmetic sequence is a series that sums all of the terms in the sequence. When the terms are put together and the total is divided by the total number of terms, the result is the arithmetic mean or the mean of the sum of the terms. The arithmetic mean may be calculated using the formula text = frac n text. The observation that when the first and last terms of an arithmetic sequence are added, the sum is the same as when the second and next to last terms are added, or when the third and third to last terms are added, provides a quick method of calculating the mean of an arithmetic sequence.
The mean of an arithmetic sequence is calculated by dividing the total by the number of terms in the sequence; hence, the mean of an arithmetic sequence is half the sum of the first and final terms.
Instead, by restricting the total to a specific number of items, it is possible to find the mean of a partial sum.
Other Types of Sequences
Observations from experiments or measurements of natural occurrences are frequently used to create numerical sequences. Such sequences can be made up of random numbers, although they are more typically made up of arithmetic or other ordered lists of numbers than random numbers. Geometric sequences, as opposed to arithmetic sequences, vary in that they share a common component rather than a common difference in their composition. To avoid the repetition of the same number being added or deleted for each new phrase, a number is multiplied or divided for each new term that is added.
Other sequences are governed by whole distinct sets of laws.
The numbers are as follows: 1, 1, 2, 3, 5, 8, and so on.
Arithmetic sequences are straightforward, yet they have a variety of practical applications.
Arithmetic Sequences and Series – MathBitsNotebook(A2
Some sequences are composed of simply random values, while others have a definite pattern that is used to arrive at the sequence’s terms. Thearithmetic sequence(or progression),for example, is based upon the addition of a constant value to arrive at the next term in the sequence. Arithmetic sequences follow a pattern ofadding a fixed amount from one term to the next. The number being added to each term is constant (always the same).a1,(a1+ d),(a1+ 2d),(a1+ 3d),.

When arithmetic sequences are graphed, they exhibit an alinear character (as a scatter plot). The domain of the sequence is represented by the counting numbers 1, 2, 3, 4, and so on (representing the position of each term), while the range of the sequence is represented by the actual terms of the series. While the xaxis rises by a constant value of one in the graph shown above, the yaxis increases by a constant value of three in the graph displayed above. Arithmetic Sequences are a type of mathematical sequence.
Arithmetic Sequence:  Common Difference,d: 
1, 6, 11, 16, 21, 26,.  d= 5.A 5 isaddedto each term to arrive at the next term.OR. thedifferencea2a1= 5. 
10, 8, 6, 4, 2, 0, 2, 4,.  d= 2.A 2 isaddedto each term to arrive at the next term.OR. thedifferencea2a1= 2. 
d= ½.A ½ isaddedto each term to arrive at the next term.OR. thedifferencea2a1= ½. 
When the terms of a sequence areadded together, the sum is referred to as aseries.We will be working withfinite sums(the sum of a specific number of terms). This is the sum of the firstnterms.
S n=a1+(a1+ d)+(a1+2 d)+(a1+3 d)+(a1+4 d)+(a1+5 d)+(a1+6 d)+(a1+7 d)+(a1+8 d)+(a1+9 d)+(a1+(n 1) d)+(a1+(n 1) d)+(a When the terms of an arithmetic sequence are added together, the result is called an anarithmetic series. Formulas that are used in conjunction with arithmetic sequences and arithmetic series include:
Tofind any term of anarithmetic sequence:wherea1is the first term of the sequence, dis the common difference,nis the number of the term to find. Note:you may seea1simply referred to asa.  To find thesum of a certain number of termsof anarithmetic sequence:whereSnis the sum ofnterms (nthpartial sum),a1is the first term,a nis thenthterm. Note:(a1+a n)/2 is the mean (average) of the first and last terms. The sum can be thought of as thenumber of terms times the average of the first and last terms. This formula may also appear as 
To learn more about “How These Formulas Were Created,” please visit this page. Examine a number of examples that use arithmetic sequences and series to demonstrate their use. Read the “Answers” attentively in order to obtain ” suggestions ” on how to cope with the questions in this section.
Questions:  Answers: 
1.Find the common difference for this arithmetic sequence: 4, 15, 26, 37,.  The common difference,d,can be found bysubtracting the first term from the second term, which in this example yields 11. Checking shows that 11 is the difference between all of the terms. 
2.Find the common difference for the arithmetic sequence whose formula is:a n= 6 n+ 3.  A listing of the terms will show what is happening in the sequence (start withn= 1). 9, 15, 21, 26, 33,.The common difference is 6. 
3.Find the 10 thterm of the sequence:3, 5, 7, 9, 11,.  By observation_n= 10,a1= 3,d= 2Use the formula for thenthterm.The 10 thterm is 21. 
4.Finda7for an arithmetic sequence where_a1= 3 xandd = x.  By observation_n= 7,a1= 3 x,d= – xYour answer will be in terms of x. 
5.Given the arithmetic sequence:f(1) = 4;f(n) =f(n 1) + 3.Findf(5).  Don’t let the change tofunctional notationdistract you. This problem showsrecursive form:each term is defined by the term immediately in front of it.The first term is 4 and the common difference is 3. Since we only need the fifth term, we can get the answer by observation: 4, 7, 10, 13, 16f(5) = 16 
6.Findt15for an arithmetic sequence where_t3= 4 + 5i andt6= 13 + 11 iNOTE:Using high subscript – low subscript + 1 will count the number of terms.  Notice the change of labeling fromatot.The letter used in labeling is of no importance. Let’s get a visual of this problem. Using the third terms as the “first” term, find the common difference from these known terms.Now, fromt3tot15is 13 terms. t15= 4 + 5 i+ (131)(3 +2 i) = 4 + 5 i36 +24 i = 40 + 29 i 
7.Find an explicit formula and a recursive formula for the sequence:1, 3, 5, 7, 9,.  Theexplicit formulaneeds to relate the subscript number of each term to the actual value of the term. These terms are odd numbers (a good formula pattern to remember).a n= 2 n 1Substitutingn= 1 gives 1.Substitutionn =2 gives 3, and so on.Therecursive formula,where each term is based upon the term immediately in front of it, is easy to find since the common difference is 2.a1= 1a n= an 1+ 2. 
8.The first three terms of an arithmetic sequence are represented byx+ 5, 3 x+ 2, and 4 x+ 3 respectively. Find the numerical value of the 10 thterm of this sequence.  Represent the common difference between the terms:(3 x+ 2) – (x+ 5) = 2 x 3 (the common difference)(4 x+ 3) – (3 x+ 2) =x+ 1 (the common difference)Since the common difference must be constant, weset these values equal and solve forx.2 x 3 =x+ 1 x= 4 The sequence is 9, 14, 19,., common difference of 5.The 10 thterm = 9 + (10 – 1)(5) =54 
9.Find the sum of the first 12 positive even integers.Notice how BOTH formulas work together to arriveat the answer.  The word “sum” indicates the need for the sum formula.positive even integers: 2, 4, 6, 8,. n= 12,a1= 2,d= 2We are missinga12, for the sum formula, so we use the “any term” formula to find it.Now, we use this information to find the sum: 
10.Insert 3 arithmetic means between 7 and 23.Note:In this context, anarithmetic meanis the term between any two terms of an arithmetic sequence. It is simply the average (mean) of its surrounding terms.  While there are several solution methods, we will use our arithmetic sequence formulas. Draw a picture to better understand the situation.7, _, _, _, 23This set of terms is an arithmetic sequence.We know the first term,a1, the last term,a n, but not the common difference,d.This question gives NO indicationof “sum”, so avoid that formula. Find the common difference:Now, insert the terms usingd: 7,11, 15, 18, 23 
11.In an arithmetic sequence,a4= 19 and a7= 31. Determine the formula fora n, thenthterm of this sequence.  Visualize the problem by modeling the terms from the fourth to the seventh.19, _. _, 31Temporarily imagine that 19 is the first term.This will allow us to find the common difference.Imagined Observations_a1= 19,a 4= 31,n= 4. 
12.Find the number of terms in the sequence:7, 10, 13,., 55Note:nmustbe an integer!  By observation_a1= 7,a n= 55,d= 3. We need tofindn. This question makes No mention of “sum”, so avoid that formula.When solving forn,be sure your answer is a positiveinteger.There is no such thing as a fractional number of terms in a sequence! 
13.A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater?  The seating pattern is forming an arithmetic sequence:60, 68, 76,.We need to find the “sum” of all of the seats. By observation_n= 20,a1= 60,d= 8 and we needa20for the sum.Now, use the sum formula.There are 2720 seats. 
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