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General Term of an Arithmetic Sequence

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Example: Finding a Term Given Another Term and the Common Difference

Find the twelfth term of an arithmetic sequence whose seventh term is 10 and whose common difference is $-2$ , and give the general term formula.

Step 1 — Find the first term. Because the first term $a_1$ is unknown, use the general term formula with the information about the seventh term: $a_n = a_1 + (n - 1)d$ . Substitute $a_7 = 10$ , $n = 7$ , and $d = -2$ :

$10 = a_1 + (7 - 1)(-2)$

Simplify inside the parentheses and multiply:

$10 = a_1 + (6)(-2)$

$10 = a_1 - 12$

Add 12 to both sides to isolate $a_1$ :

$a_1 = 22$

Step 2 — Find the twelfth term. Now substitute $a_1 = 22$ , $n = 12$ , and $d = -2$ into the formula:

$a_{12} = 22 + (12 - 1)(-2)$

$a_{12} = 22 + (11)(-2)$

$a_{12} = 22 - 22 = 0$

The twelfth term is 0.

Step 3 — Write the general term. Substitute $a_1 = 22$ and $d = -2$ into the formula and simplify:

$a_n = 22 + (n - 1)(-2)$

$a_n = 22 - 2n + 2$

The general term is $a_n = -2n + 24$ .

This example illustrates a two-stage application of the general term formula: first working backward from a known term to recover $a_1$ , then using $a_1$ to compute any other term or to write the explicit formula for the sequence.

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Updated 2026-05-26

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OpenStax Intermediate Algebra

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