# What Is An Example Of An Arithmetic Sequence? (Question)

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.

## 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 an in an arithmetic sequence?

Definition: An arithmetic sequence is a sequence of the form a, a + d, a + 2d, a + 3d, a + 4d, … The number a is the first term, and d is the common difference of the. sequence. The nth term of an arithmetic sequence is given by. an = a + (n – 1)d.

## What are 2 examples of arithmetic sequences in real life?

Examples of Real-Life Arithmetic Sequences

• Stacking cups, chairs, bowls etc.
• Pyramid-like patterns, where objects are increasing or decreasing in a constant manner.
• Filling something is another good example.
• Seating around tables.
• Fencing and perimeter examples are always nice.

## How do you know if a sequence is arithmetic?

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 type of sequence is 80 40 20?

This is a geometric sequence since there is a common ratio between each term.

## How do you find arithmetic?

An arithmetic sequence is defined as a series of numbers, in which each term (number) is obtained by adding a fixed number to its preceding term. Sum of arithmetic terms = n/2[2a + (n – 1)d], where ‘a’ is the first term, ‘d’ is the common difference between two numbers, and ‘n’ is the number of terms.

## How is arithmetic sequence used in daily life?

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.

## How do you use arithmetic sequence in real life?

For example, when you are waiting for a bus. Assuming that the traffic is moving at a constant speed you can predict the when the next bus will come. If you ride a taxi, this also has an arithmetic sequence. Once you ride a taxi you will be charge an initial rate and then a per mile or per kilometer charge.

## How do you use arithmetic mean in real life?

The arithmetic mean is used frequently not only in mathematics and statistics but also in fields such as economics, sociology, and history. For example, per capita income is the arithmetic mean income of a nation’s population.

## What is not an arithmetic sequence?

The following are not examples of arithmetic sequences: 1.) 2,4,8,16 is not because the difference between first and second term is 2, but the difference between second and third term is 4, and the difference between third and fourth term is 8. No common difference so it is not an arithmetic sequence.

## What is the common difference arithmetic sequence?

In arithmetic sequences, the common difference is simply the value that is added to each term to produce the next term of the sequence.

## 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:

The difference between one term and the next is a constant in an Arithmetic Sequence. We just add the same amount of value each time, to put it another way. endlessly.

• And this is the first period
• And distinguish the distinction between the two concepts (known as the “common difference”)

### Example: (continued)

And this is the first term; explain how the concepts vary (this is referred to as the “common difference”); and

• 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 five-point 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:

In addition, the following values are shown below and above it: “Sum upnwherengoes from 1 to 4,” the message states. 10 is the answer It can be used in the following ways:

• 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 + (n-2)d) + (a + (n-1)d) S = (a + (n-1)d) + (a + (n-2)d) + . + (a + d) + a 2S = (2a + (n-1)d) + (2a + (n-1)d) + . + (2a + (n-1)d) + (2a + (n-1)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:

## 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

In such case, the thterm of an arithmetic series of the form A1, A2, A3,. is given byan = a1 + (n-1) 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.

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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 n-th phrase, a = (a + 1)d
• The sum of n terms, Sn =/2 (or) n/2 (2a + 1)d
• The n-th term, a = (a + 1)d
• The n-th term, a = a + (n-1)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 n-th 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 Sequence-Related 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

1. 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) d-248 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?

It is known as an arithmetic sequence when the difference between every two successive terms in a series is the same. For example, the numbers 3, 8, 13, and 18 are arithmetic because they are consecutive numbers.

### 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?

In an arithmetic series, Sn represents the sum of the first n terms, with the first term indicated by a, and the common difference denoted by d. There are two formulae for calculating Sn.

### 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?

The number that appears in the first position from the left in an arithmetic sequence is referred to as the first term. A ‘a’ is used to signify that it is present. If the letter ‘a’ is not provided in the issue, then the problem may contain some information about d (or) a (or) Sn. In order to solve for ‘a’, we will simply insert the supplied values into the an or Sn formulae.

### 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:

• For arithmetic sequences, the following are the stages to be followed:

### 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)

## 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. There is no common difference between the numbers, hence it is not an arithmetic sequence.

## 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 12-9-6-3-0-3-6-0 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.
• 3: The number series 2,3,5,8,12,17,23,.
• Differencea2 is 1, but the following differencea3 is 2, and the differencea4 is 3.
• Geometric sequences are another type of sequence.
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## Arithmetic Sequence: Formula & Definition – Video & Lesson Transcript

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. Thenthterm in an arithmetic series may be represented by the formula forman=dn+c, where d is the common difference. As soon as you’ve determined the common difference, you can calculate the value ofcby substituting 1 fornand the first term in the series fora1 into the equation. 1, 5, 9, 13, 17, 21, 25, is an arithmetic series having a common difference of 4, as seen in Example 1: 1, 5, 9, 13, 17, 21, 25,.

1. So, in order to get a formula for the thenthterm, we replace n = 1, a1 = 1, andd = 4 in an= dn + c to findc.
2. As an example, the arithmetic sequence 12-9-6-3-0-3-6) is an arithmetic series having a common difference of three.
3. It is important to note that, because the series is decreasing, the common difference is a negative number.
4. Thenthterm is found by substitutingn=1, a1=12, and d=3inan=dn+c to get the formula n=1, a1=12, and d=3inan=dn+c to get the formula c.12 = 3(1) plus cc = 15.
5. As an illustration, consider the third case.

In this case, the first difference is 1, and the second is 2, but the third is also 2. Using this sequence, it is impossible to write a formula in the form of forman=dn+c. Geometric sequences are another term for this.

## 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 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 time-consuming and inefficient.

## 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

The succession of numbers in anarithmetic A series of numbers in which each succeeding number is the sum of the preceding number plus certain constants, for example, the development of numbers in arithmetic terms 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., 2). an=an−1+d Number Sequences in Arithmetic As a result, the constant is referred to as the common difference since anan1=d.

An arithmetic sequence is, for example, the series of positive odd integers: 1, 3, 5, 7, 9,.

### 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. It is possible that the initial term of an arithmetic series will not be provided in some instances.

### 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!

## 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 time-consuming.

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.

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. Sn=n(a1+an)2S50=50.(a1+a50)2=50(4+249) 2=25(253)=6,325 Answer_S50=6,325

### 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,.

### 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

1. 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
2. D=3
3. A1=12
4. D=2
5. A1=15
6. D=5
7. A1=7
8. D=4
9. D=1
10. A1=23
11. D=13
12. A 1=1
13. D=12
14. A1=54
15. D=14
16. A1=1.8
17. D=0.6
18. A1=4.3
19. D=2.1
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,.
2. 4.4, 7.5, 13.7, 16.8,.
3. 3, 8, 13, 18, 23,.
4. 3, 7, 11, 15, 19,.
5. 6, 14, 22, 30, 38,.
6. 5, 10, 15, 20, 25,.
7. 2, 4, 6, 8, 10,.
8. 12,52,92,132,.
9. 13, 23, 53,83,.
10. 14,12,54,2,114,. Find the positive odd integer that is 50th
11. Find the positive even integer that is 50th
12. Find the 40 th term in the sequence that consists of every other positive odd integer in the following format: 1, 5, 9, 13,.
13. Find the 40th term in the sequence that consists of every other positive even integer: 1, 5, 9, 13,.
14. Find the 40th term in the sequence that consists of every other positive even integer: 2, 6, 10, 14,.
15. 2, 6, 10, 14,. What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
16. What number is the phrase 172 in the arithmetic sequence 4, 4, 12, 20, 28,.
17. What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
18. 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
19. Find an equation that yields the general term in terms ofa1and the common differenced, given the arithmetic sequence described by the recurrence relationan=an1-9wherea1=4 andn1
20. This is the problem.
1. Locate a formula for the general term in the arithmetic series and apply it to identify the 100th term
2. Given the arithmetic sequence 3, 9, 15, 21, 27,.
3. 3, 8, 13, 18, 23,.
4. 3, 7, 11, 15, 19,.
5. 6, 14, 22, 30, 38,.
6. 5, 10, 15, 20, 25,.
7. 2, 4, 6, 8, 10,.
8. 12,52,92,132,172,.
9. 13,23,53,83,113,.
10. 0.8, 2, 3.2, 4.4, 5.6,.
11. 4.4, 7.5, 10.6, 13.7, 16.8,.
12. 4.4, Find the positive odd integer that is 50th
13. Find the positive even integer that is 50th
14. And so on. Find the 40th term in the series that consists of every other positive odd integer in the following format: the first five terms in a series consisting of every other positive even number are 1, 5, 9, 13,.
15. Find the fortyth term in a sequence consisting of every other positive even integer are 1, 5, 9, 13,.
16. Numbers 2 through 6 and 10, 14, and so on When mathematical sequences 15 and 5 are used, what number is the term 355 in the sequence? When arithmetic sequences 4 and 4 are used, what number is the phrase 172? 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
17. Using the arithmetic sequence described by the recurrence relationan=an1-9wherea1=4 andn1 and the common differenced, find an equation that provides the general term in terms of a1 and the common differenced.
1. Find all possible arithmetic means between the given terms: a1=3anda6=17
2. A1=5anda5=7
3. A2=4anda8=7
4. A5=12anda9=72
5. A5=15anda7=21
6. A6=4anda11=1
7. A7=4anda11=1
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### Part B: Arithmetic Series

1. Make a calculation for the provided total based on the formula for the general term an=3n+5
2. S100
3. An=5n11
4. An=12n
5. S70
6. An=132n
7. S120
8. An=12n34
9. S20
10. An=n35
11. S150
12. An=455n
13. S65
14. An=2n48
15. S95
16. An=4.41.6n
17. S75
18. An=6.5n3.3
19. S67
20. An=3n+5
1. Consider the following values: n=1160(3n)
2. N=1121(2n)
3. N=1250(4n3)
4. N=1120(2n+12)
5. N=170(198n)
6. N=1220(5n)
7. N=160(5212n)
8. N=151(38n+14)
9. N=1120(1.5n2.6)
10. N=1175(0.2n1.6)
11. 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.
1. 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
2. The sum of the first 200 positive odd integers
3. The sum of the first 50 positive even integers
4. The sum of the first 200 positive even integers
5. The sum of the first 100 positive even integers
6. The sum of the firstk positive odd integers
7. The sum of the firstk positive odd integers the sum of the firstk positive even integers
8. The sum of the firstk positive odd integers
9. 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
10. 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 10-year 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 ten-year 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

1. 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
2. Discuss strategies for computing sums in situations when the index does not begin with one. For example, n=1535(3n+4)=1,659
3. N=1535(3n+4)=1,659
4. Carl Friedrich Gauss is the subject of a well-known 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?

1. Is the Fibonacci sequence an arithmetic series, or is it a mathematical 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 certain situations? Make a personal example to illustrate your point
2. Discuss strategies for computing sums in situations when the index does not begin at one (1). n=1535(3n+4)=1,659 is an example of the number n=1535(3n+4)=1,659 Carl Friedrich Gauss was once accused of misbehaving at school, according to a well-known legend. As a punishment, his instructor assigned him the chore of adding the first 100 integers to his list of disciplinary measures. Apparently, Gauss responded accurately within seconds, according to mythology. In what way do you believe he was able to come up with the solution so rapidly, and how do you think he did it?
1. 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

## Examples of Real-Life Arithmetic Sequences

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

## Sequences and Series

SEQUENCES AND SERIESUnit Overview Animportant mathematical skill is discovering patterns. In this unit you willinvestigate different types of patterns represented in sequences. You will thenuse the Binomial Theorem to expand powers of binomials.Arithmetic SequencesA sequence is a list ofnumbers in a particular order. Each number in a sequence is called a term.The first term is represented bya1, the second term isrepresented bya2, and so on.a nrepresents the nth term of the sequence.One kind of sequence is anarithmeticsequence.

1. For instance, inthe sequence 2, 7, 12, 17, there are 4 terms in the sequence thereforen= 4 and the commondifference isd= 5.
2. When there arenterms in a sequence, the commondifferencedis added to each of the terms in the sequencen– 1 times.
4. Following is an illustration of a GeometricSequence.
5. Another way to look at this sequence is as follows: a1= 2, a2=a1 2, a3=a2 2, a4=a3 2, a5=a4 2.
6. Whenever a Geometric Sequence is used, a common ratio is defined as the integer that multiplies the current term to provide the next result.
7. In this case, the commonratio is three, which is expressed by the symbol r= three.

To describe the terms of ageometric sequence in terms of ageometric sequence, the following formulas can be used: a1, a2=A1r, a3=A2r, a4=A3r, a5=A4r.

If the sequence is arithmetic, indicate the common difference between the two numbers.

Identify the sequences in each group.

1, 4, 9, 16,.

The same number is neither added or multiplied more than once to generate the following term, nor is it subtracted from the previous term.

25, 50, 75, 100, and so on In mathematics, d= 25 To double-check the answer, click on “Click here.” Identify the sequences in each group.

Return to the previous part and answer the questions 6-11 before moving on to the next segment of the exam.

A sequence is a mathematical phrase that represents the sum of the terms in a sequence.

The indicated sum of the terms of an arithmetic sequence is referred to as the anarithmetic series. Consider the numbers 6, 9, 12, 15, 18, and so on. It is referred to as S5 when the sum of the first five terms of this sequence is considered.

Consider the following series for the purpose of developing a formula for the sum of the firstnterms of a series. Series and Sequences of Arithmetic Operations – Amphitheater (03:08) Stop! Return to the previous part and answer the questions 12-17 before moving on to the next segment of the exam. Geometric Series is a series of geometric figures. Ageometricseriesis the sum of the terms of a geometric sequence that has been indicated. Consider the numbers 3, 9, 27, 81, and so on. The sum of the first five terms is indicated by the symbol S5is: Geometric Sequences and Series – FamilyTree Graphing Software (02:49) Stop!

• Word Sequences and series problems are addressed in this section.
• How many bacteria will be present after 24 hours if there are 200 germs present at the start of the experiment?
• The following sequence is produced by starting with 200 and doubling it.
• Step 2: Identify the factors that are involved.
• As a result, we’re seeking for the eighth word in the series.
• Make use of the geometric sequence formula to create a sequence.
• how many direct ancestors would you have if youcould trace your family back for 10 generations?Step1:Determine whether the situation represents anarithmetic sequence, geometric sequence, arithmetic series or geometric series.The sequence 2, 4, 8, … represents a geometricsequence.
• Goingback 10 generations, a person would have 2,046 direct ancestors.
• For eachadditional car Justin sells during the month, he receives \$10.00 more on hiscommission.
• Commissionon 1 stcar – \$150Commissionon2 ndcar – \$160Commissionon 3 rdcar – \$170Andso on.The150, 160, 170, … represents an arithmetic sequence.The question asks for the total amount ofmoney earned so use the formula for an arithmetic series.Step3:Substituteand evaluate.

Thecommission earned on the 16th car sold is \$300.Nowsubstitute in the formula for an arithmetic series to determine Justin’scommission for the month.Justin earns \$3,600 for the month Justinearns \$3,600 for the month.Stop!Go to Questions23-26 about thissection, then return to continue on to the next section.BinomialTheoremRecallfrom algebra that a binomial is an expression of the form: Inprevious units, you have learned how to expand binomials wheren= 2 orn= 3.However, expanding a binomial for large values ofnwould be quite time consuming.

Fortunately, there is a theorem and a devicethat gives a pattern for expanding binomials for any value ofn.The theorem is called the Binomial Theoremand the device applied to this theorem is called Pascal’s triangle.BinomialTheorem:Ifnis a positive integer, thenNotice in this expression that the exponent onxdecreasesby one for each new term while the exponent onbincreases by one foreach new term.Connected to theBinomial Theoremis the famous“Pascal’s Triangle”, which is given below and can be used to find thecoefficients of a binomial expansion.Notice how the numbers in the row above determines thenumbers in each row of the triangle.

For example the number “2” in row3 is thesum of 1 + 1 in row2.

Example1:Use Pascal’sTriangle to expand the following binomial.Step1:From Pascal’sTriangle,n= 4, the exponent,corresponds to the numbersfound in row five.These numbers are; 1 4 6 4 1.Step2:Apply thesenumbers as the coefficients of the terms of the expanded polynomial.

Example2:UsePascal’s Triangle to expand the following binomial.Step1:From Pascal’sTriangle,n= 6, the exponent,corresponds to the numbers found in row seven.These numbers are: 1 6 15 20 15 6 1Step2:Apply thesenumbers as the coefficients of the terms of the expanded polynomial and applydecreasing exponents on the entire first term,”5 a ” andincreasing exponentson the entire 2nd term, “2 b ”as follows.We will not apply the exponents toeach term as it is obvious that very large numbers will result quite fast.

For examplethe third term in the answer if expanded will be 37500 a4b2.Stop!Go to Questions27-34 to complete thisunit.