Write a general formula for the arithmetic sequence given. (B) 1 2 , 5 6 , 7 6 , 3 2 , … \frac12,\frac56,\frac76,\frac32,\ldots

a n = 1 3 n + 1 6 a_{n}=\frac13n+\frac16

a n = 1 3 n + 5 6 a_{n}=\frac13n+\frac56

a n = − 1 3 n + 5 6 a_{n}=-\frac13n+\frac56

a n = 1 3 n + 2 6 a_{n}=\frac13n+\frac26

15. Sequences, Series, and the Binomial Theorem

Arithmetic Sequences

15. Sequences, Series, and the Binomial Theorem

Arithmetic Sequences: Videos & Practice Problems

Video Lessons Worksheet

Topic summary

An arithmetic sequence is defined by a constant common difference $d$ between consecutive terms. The general term $a_{n}$ of an arithmetic sequence can be found using the formula $a_{n} = a_{1} + (n - 1) \times d$ , where $a_{1}$ is the first term. This formula enables efficient calculation of any term, such as the twentieth term, without listing all terms. Understanding how to derive $d$ and apply this formula is essential for analyzing sequences and their patterns.

concept

Intro to Arithmetic Sequences

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Intro to Arithmetic Sequences Video Summary

An arithmetic sequence is a type of sequence where each term changes by the same fixed amount, known as the common difference. This common difference, represented by the letter d, is the difference between any two consecutive terms in the sequence. For example, in the sequence 2, 6, 10, 14, the common difference is 4 because each term increases by 4 from the previous term. This consistent change defines the sequence as arithmetic.

To find the common difference, you subtract the first term from the second term, or generally, subtract any term from the term that follows it. Mathematically, this is expressed as:

\[d = a_{n+1} - a_n\]

where $a_n$ is the nth term and $a_{n+1}$ is the term immediately after it.

Once the common difference is known, you can determine subsequent terms by adding (or subtracting, if the common difference is negative) this value to the previous term. For instance, if the first term $a_1$ is 10 and the second term $a_2$ is 8, the common difference is:

\[d = 8 - 10 = -2\]

This negative common difference indicates the sequence decreases by 2 each time. Therefore, the third term $a_3$ is:

\[a_3 = a_2 + d = 8 + (-2) = 6\]

and the fourth term $a_4$ is:

\[a_4 = a_3 + d = 6 + (-2) = 4\]

Understanding arithmetic sequences and the role of the common difference is fundamental for analyzing patterns and predicting future terms in a sequence. This concept lays the groundwork for more advanced topics involving sequences and series.

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Follow along with each video using our printable worksheets

Problem

Find the common difference of the following sequence.
(A) $- 9, - 4, 1, 6, \dots$

$4$

$negative 4$

$negative 5$

$5$

Problem

Find the common difference of the following sequence.
(B) $\frac{7}{4}, \frac{5}{4}, \frac{3}{4}, \frac{1}{4}, \dots$

$d = - 2$

$d = - \frac{2}{4}$

$2$

$\frac{2}{4}$

Problem

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.
(A) $a_{1} = - 12; d = - 5$

$- 12, - 17, - 22, - 27, - 32$

$- 32, - 27, - 22, - 17, - 12$

$- 12, - 7, - 2, 3, 8$

$8, 3, - 2, - 7, - 12$

Problem

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.
(B) $a_{1} = \frac{3}{2}, d = \frac{1}{4}$

$three halves comma 1 comma one half comma 0 comma negative one half$

$three halves comma five fourths comma 1 comma three fourths comma one half$

$three halves comma seven fourths comma 2 comma nine fourths comma five halves$

$three halves comma 2 comma five halves comma 3 comma seven halves$

concept

General Term of an Arithmetic Sequences

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General Term of an Arithmetic Sequences Video Summary

An arithmetic sequence is defined by a constant difference, known as the common difference d, between consecutive terms. To find the general term, or the nth term a_n, of an arithmetic sequence, you use the formula:

\[a_n = a_1 + (n - 1)d\]

Here, a₁ represents the first term of the sequence, and d is the common difference. This formula allows you to calculate any term in the sequence without listing all the previous terms.

For example, consider the arithmetic sequence 2, 6, 10, 14, ... where the common difference d is 4 (since each term increases by 4). The first term a₁ is 2. Plugging these values into the formula gives:

\[a_n = 2 + (n - 1) \times 4\]

This can be simplified by distributing the 4:

\[a_n = 2 + 4n - 4 = 4n - 2\]

Both forms are correct, but depending on the context, you may be asked to leave the formula simplified or in its original form.

Using this formula, you can quickly find any term in the sequence. For instance, to find the 20th term a₂₀, substitute n = 20:

\[a_{20} = 2 + (20 - 1) \times 4 = 2 + 19 \times 4 = 2 + 76 = 78\]

This method is much faster than adding the common difference repeatedly.

Sometimes, you may be given the first term and the common difference directly, allowing you to write the general term formula immediately. For example, if a₁ = 8 and d = -6, then:

\[a_n = 8 + (n - 1)(-6)\]

Other times, you might need to find the common difference using additional information. Suppose you know the first term a₁ = 2 and the fifth term a₅ = 14. Using the general term formula:

\[a_5 = a_1 + (5 - 1)d\]

Substitute the known values:

\[14 = 2 + 4d\]

Solving for d:

\[14 - 2 = 4d \implies 12 = 4d \implies d = 3\]

Now that the common difference is found, the general term formula becomes:

\[a_n = 2 + (n - 1) \times 3\]

This approach highlights the importance of understanding how the position of a term relates to the first term and the common difference. By mastering this formula and the process of identifying a₁ and d, you can efficiently analyze and work with arithmetic sequences in various mathematical contexts.

Problem

Write a general formula for the arithmetic sequence given.
(A) $- 5, - 12, - 19, - 26, \dots$

$a_{n} = 7 n - 12$

$a_{n} = - 7 n + 2$

$a_{n} = 7 n + 2$

$a_{n} = - 7 n - 12$

Problem

Write a general formula for the arithmetic sequence given.
(B) $\frac{1}{2}, \frac{5}{6}, \frac{7}{6}, \frac{3}{2}, \dots$

$a_{n} = \frac{1}{3} n + \frac{5}{6}$

$a_{n} = - \frac{1}{3} n + \frac{5}{6}$

$a_{n} = \frac{1}{3} n + \frac{2}{6}$

$a_{n} = \frac{1}{3} n + \frac{1}{6}$

Problem

Find the indicated term of each arithmetic sequence.
(A) $a_{n} = 7 + 3 (n - 1)$
Find $a sub 12$ .

$40$

$43$

$37$

$34$

Problem

Find the indicated term of each arithmetic sequence.
(B) $a_{n} = \frac{5}{2} + \frac{3 (n - 1)}{4}$
Find $a sub 6$ .

$25 over 4$

$5$

$10$

$25 over 2$

example

General Term of an Arithmetic Sequences Example 1

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General Term of an Arithmetic Sequences Example 1 Video Summary

In an arithmetic sequence, each term is found by adding a constant value called the common difference to the previous term. Given the fifth term as -12 and the fifteenth term as 18, we can determine the common difference and then find the ninth term. The key is to recognize that the difference in term positions corresponds to how many times the common difference is added. Specifically, moving from the fifth term to the fifteenth term involves adding the common difference 10 times, since 15 - 5 = 10.

Using the formula for the nth term of an arithmetic sequence, $a_n = a_m + (n - m)d$, where $a_n$ is the nth term, $a_m$ is the mth term, and $d$ is the common difference, we substitute the known values:

\$18 = -12 + 10d\(

Solving for \)d\(, add 12 to both sides:

\)18 + 12 = 10d\(

\)30 = 10d\(

Divide both sides by 10:

\)d = 3\(

With the common difference found, we can now calculate the ninth term. Since the ninth term is four positions after the fifth term (9 - 5 = 4), we add the common difference four times to the fifth term:

\)a_9 = a_5 + 4d = -12 + 4(3) = -12 + 12 = 0$

Therefore, the ninth term of the arithmetic sequence is 0. This approach highlights how understanding the relationship between term positions and the common difference allows for efficient calculation of any term in an arithmetic sequence.

example

General Term of an Arithmetic Sequences Example 2

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General Term of an Arithmetic Sequences Example 2 Video Summary

To determine the number of terms in an arithmetic sequence, consider the sequence 4, 9, 14, 19, ..., with the last term given as 94. Since the number of terms between 19 and 94 is unknown, counting directly is not feasible. Instead, use the general formula for the nth term of an arithmetic sequence:

\[a_n = a_1 + (n - 1)d\]

Here, $a_n$ represents the nth term, $a_1$ is the first term, $d$ is the common difference, and $n$ is the number of terms. For this sequence, the first term $a_1$ is 4, and the common difference $d$ is 5, since each term increases by 5 (from 4 to 9, 9 to 14, and so on).

Substituting these values into the formula and setting $a_n = 94$ gives:

\[94 = 4 + (n - 1) \times 5\]

To solve for $n$, first subtract 4 from both sides:

\[94 - 4 = (n - 1) \times 5\]

\[90 = 5(n - 1)\]

Next, divide both sides by 5:

\[\frac{90}{5} = n - 1\]

\[18 = n - 1\]

Finally, add 1 to both sides to isolate $n$:

\[18 + 1 = n\]

\[n = 19\]

Therefore, the arithmetic sequence contains 19 terms. Understanding how to apply the nth term formula allows for efficient calculation of the number of terms in any arithmetic sequence when the first term, common difference, and last term are known.

example

General Term of an Arithmetic Sequences Example 3

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General Term of an Arithmetic Sequences Example 3 Video Summary

In this problem, ticket sales for a school play form an arithmetic sequence, where the number of tickets sold increases by a constant amount each night. The initial ticket sales on opening night are 120, which represents the first term of the sequence, denoted as a₁. The common difference, d, is 25, indicating that each subsequent night sells 25 more tickets than the previous night.

To find the number of tickets sold on the tenth night, we use the formula for the n-th term of an arithmetic sequence:

\[a_n = a_1 + (n - 1)d\]

Substituting the known values, where a₁ = 120, d = 25, and n = 10, we calculate:

\[a_{10} = 120 + (10 - 1) \times 25 = 120 + 9 \times 25 = 120 + 225 = 345\]

This means that on the tenth night, 345 tickets were sold. Understanding arithmetic sequences and how to apply their formulas allows us to model and predict patterns in real-world scenarios such as ticket sales growth over time.

Here’s what students ask on this topic:

An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference, denoted as $d$ . To identify an arithmetic sequence, check if the difference between each pair of consecutive terms is the same. For example, in the sequence 2, 6, 10, 14, the difference between terms is always 4, so it is an arithmetic sequence with $d =4$ . If the difference is negative, like in 10, 8, 6, 4, the common difference is negative ( $d =-2$ ), but it is still arithmetic. Recognizing this pattern is essential for working with arithmetic sequences.

To find the common difference $d$ in an arithmetic sequence, subtract any term from the term that follows it. Mathematically, $= a_{n+1} - a_n$ , where $a_n$ and $a_{n+1}$ are consecutive terms. For example, if the sequence is 10, 8, 6, 4, then $d = 8 - 10 = -2$ . This means each term decreases by 2. You can use any two consecutive terms to find $d$ , and it will always be the same for an arithmetic sequence.

The formula for the nth term $a_n$ of an arithmetic sequence is $a_n = a_1 + (n - 1) imes d$ , where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number. This formula allows you to find any term in the sequence without listing all previous terms. For example, if $a_1 = 2$ and $d = 4$ , then the 20th term is $a_{20} = 2 + (20 - 1) imes 4 = 2 + 76 = 78$ . This is a powerful tool for quickly calculating terms in arithmetic sequences.

If you know the first term $a_1$ and another term $a_k$ at position $k$ , you can find the common difference $d$ using the formula derived from the nth term: $a_k = a_1 + (k - 1) imes d$ . Rearranging to solve for $d$ , you get $d = \frac{a_k - a_1}{k - 1}$ . For example, if $a_1 = 2$ and $a_5 = 14$ , then $d = \frac{14 - 2}{5 - 1} = \frac{12}{4} = 3$ . This method is useful when the common difference is not given directly.

To find a specific term in an arithmetic sequence using the general term formula, plug the term number $n$ into the formula $a_n = a_1 + (n - 1) imes d$ . For example, if the first term $a_1 = 2$ and the common difference $d = 4$ , to find the 20th term, substitute $n = 20$ : $a_{20} = 2 + (20 - 1) imes 4 = 2 + 76 = 78$ . This approach is faster than listing all terms and helps analyze sequences efficiently.

Your Beginning & Intermediate Algebra tutors

Arithmetic Sequences: Videos & Practice Problems

Intro to Arithmetic Sequences

Intro to Arithmetic Sequences Video Summary

Find the common difference of the following sequence.
(A) $- 9, - 4, 1, 6, \dots$

Find the common difference of the following sequence.
(B) $\frac{7}{4}, \frac{5}{4}, \frac{3}{4}, \frac{1}{4}, \dots$

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.
(A) $a_{1} = - 12; d = - 5$

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.
(B) $a_{1} = \frac{3}{2}, d = \frac{1}{4}$

General Term of an Arithmetic Sequences

General Term of an Arithmetic Sequences Video Summary

Write a general formula for the arithmetic sequence given.
(A) $- 5, - 12, - 19, - 26, \dots$

Write a general formula for the arithmetic sequence given.
(B) $\frac{1}{2}, \frac{5}{6}, \frac{7}{6}, \frac{3}{2}, \dots$

Find the indicated term of each arithmetic sequence.
(A) $a_{n} = 7 + 3 (n - 1)$
Find $a sub 12$ .

Find the indicated term of each arithmetic sequence.
(B) $a_{n} = \frac{5}{2} + \frac{3 (n - 1)}{4}$
Find $a sub 6$ .

General Term of an Arithmetic Sequences Example 1

General Term of an Arithmetic Sequences Example 1 Video Summary

General Term of an Arithmetic Sequences Example 2

General Term of an Arithmetic Sequences Example 2 Video Summary

General Term of an Arithmetic Sequences Example 3

General Term of an Arithmetic Sequences Example 3 Video Summary

Here’s what students ask on this topic:

What is an arithmetic sequence and how do you identify it?

How do you find the common difference in an arithmetic sequence?

What is the formula for the nth term of an arithmetic sequence?

How do you find the common difference if you know the first term and another term in the sequence?

How can you use the general term formula to find a specific term in an arithmetic sequence?

Your Beginning & Intermediate Algebra tutors

Arithmetic Sequences: Videos & Practice Problems

Intro to Arithmetic Sequences

Intro to Arithmetic Sequences Video Summary

Find the common difference of the following sequence.(A) −9,−4,1,6,…-9,-4,1,6,…

Find the common difference of the following sequence.(B) 74,54,34,14,…\frac74,\frac54,\frac34,\frac14,\ldots

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.(A) a1=−12;d=−5a_1=-12; d=-5

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.(B) a1=32,d=14a_1=\frac32,d=\frac14

General Term of an Arithmetic Sequences

General Term of an Arithmetic Sequences Video Summary

Write a general formula for the arithmetic sequence given.(A) −5,−12,−19,−26,…-5,-12,-19,-26,…

Write a general formula for the arithmetic sequence given.(B) 12,56,76,32,…\frac12,\frac56,\frac76,\frac32,\ldots

Find the indicated term of each arithmetic sequence.(A) an=7+3(n−1)a_n=7+3(n-1)Find a12a_{12}.

Find the indicated term of each arithmetic sequence.(B) an=52+3(n−1)4a_{n}=\frac52+\frac{3\left(n-1\right)}{4}Find a6a_6.

General Term of an Arithmetic Sequences Example 1

General Term of an Arithmetic Sequences Example 1 Video Summary

General Term of an Arithmetic Sequences Example 2

General Term of an Arithmetic Sequences Example 2 Video Summary

General Term of an Arithmetic Sequences Example 3

General Term of an Arithmetic Sequences Example 3 Video Summary

Here’s what students ask on this topic:

What is an arithmetic sequence and how do you identify it?

What is an arithmetic sequence and how do you identify it?

How do you find the common difference in an arithmetic sequence?

How do you find the common difference in an arithmetic sequence?

What is the formula for the nth term of an arithmetic sequence?

What is the formula for the nth term of an arithmetic sequence?

How do you find the common difference if you know the first term and another term in the sequence?

How do you find the common difference if you know the first term and another term in the sequence?

How can you use the general term formula to find a specific term in an arithmetic sequence?

How can you use the general term formula to find a specific term in an arithmetic sequence?

Your Beginning & Intermediate Algebra tutors

Find the common difference of the following sequence.
(A) $- 9, - 4, 1, 6, \dots$

Find the common difference of the following sequence.
(B) $\frac{7}{4}, \frac{5}{4}, \frac{3}{4}, \frac{1}{4}, \dots$

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.
(A) $a_{1} = - 12; d = - 5$

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.
(B) $a_{1} = \frac{3}{2}, d = \frac{1}{4}$

Write a general formula for the arithmetic sequence given.
(A) $- 5, - 12, - 19, - 26, \dots$

Write a general formula for the arithmetic sequence given.
(B) $\frac{1}{2}, \frac{5}{6}, \frac{7}{6}, \frac{3}{2}, \dots$

Find the indicated term of each arithmetic sequence.
(A) $a_{n} = 7 + 3 (n - 1)$
Find $a sub 12$ .

Find the indicated term of each arithmetic sequence.
(B) $a_{n} = \frac{5}{2} + \frac{3 (n - 1)}{4}$
Find $a sub 6$ .