12.2 Arithmetic Sequences

Before you get started, take this readiness quiz.

Evaluate $4 n - 1$ for the integers 1, 2, 3, and 4.
If you missed this problem, review the reference.

Solve the system of equations: $\begin{cases} x + y = 7 \\ 3 x + 4 y = 23 \end{cases} .$
If you missed this problem, review the reference.

If $f \left(n\right) = \frac{n}{2} \left(3 n + 5\right) ,$ find $f \left(1\right) + f \left(20\right) .$
If you missed this problem, review the reference.

Determine if each sequence is arithmetic. If so, indicate the common difference.

ⓐ $5 , 9 , 13 , 17 , 21 , 25 , \ldots$

ⓑ $4 , 9 , 12 , 17 , 20 , 25 , \ldots$

ⓒ $10 , 3 , −4 , −11 , −18 , −25 , \ldots$

Determine if each sequence is arithmetic. If so, indicate the common difference.

ⓐ $9 , 20 , 31 , 42 , 53 , 64 , \ldots$ ⓑ $12 , 6 , 0 , −6 , −12 , −18 , \ldots$ ⓒ $7 , 1 , 10 , 4 , 13 , 7 , \ldots$

Determine if each sequence is arithmetic. If so, indicate the common difference.

ⓐ $−4 , 4 , 2 , 10 , 8 , 16 , \ldots$ ⓑ $−3 , −1 , 1 , 3 , 5 , 7 , \ldots$ ⓒ $7 , 2 , −3 , −8 , −13 , −18 , \ldots$

Write the first five terms of the sequence where the first term is 5 and the common difference is $d = −6 .$

Write the first five terms of the sequence where the first term is 7 and the common difference is $d = −4 .$

Write the first five terms of the sequence where the first term is 11 and the common difference is $d = −8 .$

Find the fifteenth term of a sequence where the first term is 3 and the common difference is 6.

Find the twenty-seventh term of a sequence where the first term is 7 and the common difference is 9.

Find the eighteenth term of a sequence where the first term is 13 and the common difference is $−7$.

Find the twelfth term of a sequence where the seventh term is 10 and the common difference is $−2$. Give the formula for the general term.

Find the eleventh term of a sequence where the ninth term is 8 and the common difference is $−3 .$ Give the formula for the general term.

Find the nineteenth term of a sequence where the fifth term is 1 and the common difference is $−4 .$ Give the formula for the general term.

Find the first term and common difference of a sequence where the fourth term is 17 and the thirteenth term is 53. Give the formula for the general term.

Find the first term and common difference of a sequence where the third term is 2 and the twelfth term is $−25 .$ Give the formula for the general term.

Find the sum of the first 30 terms of the arithmetic sequence: 8, 13, 18, 23, 28, …

Find the sum of the first 30 terms of the arithmetic sequence: 5, 9, 13, 17, 21, …

Find the sum of the first 30 terms of the arithmetic sequence: 7, 10, 13, 16, 19, …

Find the sum of the first 50 terms of the arithmetic sequence whose general term is $a_{n} = 2 n - 5 .$

Find the sum of the first 50 terms of the arithmetic sequence whose general term is $a_{n} = 4 n + 3 .$

Find the sum: $\sum_{i = 1}^{30} \left(\right. 6 i - 4 \left.\right) .$

Find the sum: $\sum_{i = 1}^{35} \left(\right. 5 i - 3 \left.\right) .$

Determine if each sequence is arithmetic, and if so, indicate the common difference.

$4 , 12 , 20 , 28 , 36 , 44 , \ldots$

Determine if each sequence is arithmetic, and if so, indicate the common difference.

$−15 , −16 , 3 , 12 , 21 , 30 , \ldots$

Determine if each sequence is arithmetic, and if so, indicate the common difference.

$8 , 5 , 2 , −1 , −4 , −7 , \ldots$

Write the first five terms of each sequence with the given first term and common difference.

$a_{1} = 11$ and $d = 7$

Write the first five terms of each sequence with the given first term and common difference.

$a_{1} = −7$ and $d = 4$

Write the first five terms of each sequence with the given first term and common difference.

$a_{1} = 14$ and $d = −9$

Find the term described using the information provided.

Find the twenty-first term of a sequence where the first term is three and the common difference is eight.

Find the term described using the information provided.

Find the thirtieth term of a sequence where the first term is $−14$ and the common difference is five.

Find the term described using the information provided.

Find the sixteenth term of a sequence where the first term is 11 and the common difference is $−6 .$

Find the term described using the information provided.

Find the twentieth term of a sequence where the fifth term is $−4$ and the common difference is $−2 .$ Give the formula for the general term.

Find the term described using the information provided.

Find the eleventh term of a sequence where the third term is 19 and the common difference is five. Give the formula for the general term.

Find the term described using the information provided.

Find the eighth term of a sequence where the seventh term is $−8$ and the common difference is $−5 .$ Give the formula for the general term.

Find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

The second term is 14 and the thirteenth term is 47.

Find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

The second term is 13 and the tenth term is $−51 .$

Find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

The fourth term is $−6$ and the fifteenth term is 27.

Find the sum of the first 30 terms of each arithmetic sequence.

$11 , 14 , 17 , 20 , 23 , \ldots$

Find the sum of the first 30 terms of each arithmetic sequence.

$8 , 5 , 2 , −1 , −4 , \ldots$

Find the sum of the first 30 terms of each arithmetic sequence.

$−17 , −15 , −13 , −11 , −9 , \ldots$

Find the sum of the first 50 terms of the arithmetic sequence whose general term is given.

$a_{n} = 5 n - 1$

Find the sum of the first 50 terms of the arithmetic sequence whose general term is given.

$a_{n} = −3 n + 5$

Find each sum.

$\sum_{i = 1}^{40} \left(\right. 8 i - 7 \left.\right)$

Find each sum.

$\sum_{i = 1}^{50} \left(\right. 3 i + 6 \left.\right)$

Find each sum.

$\sum_{i = 1}^{35} \left(\right. −6 i - 2 \left.\right)$

In your own words, explain how to determine whether a sequence is arithmetic.

In your own words, explain how to find the general term of an arithmetic sequence.