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Sum of the First n Terms of an Arithmetic Sequence

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Example: Finding the Sum of the First 30 Terms of an Arithmetic Sequence

Find the sum of the first 30 terms of the arithmetic sequence $8, 13, 18, 23, 28, \dots$

To compute this sum, use the formula $S_n = \frac{n}{2}(a_1 + a_n)$ . The known values are $a_1 = 8$ , $d = 5$ , and $n = 30$ , but $a_n$ must be found first.

Use the general term formula to find $a_{30}$ :

$a_n = a_1 + (n - 1)d$

$a_{30} = 8 + (30 - 1) \cdot 5$

$a_{30} = 8 + (29) \cdot 5$

$a_{30} = 8 + 145 = 153$

Now substitute $a_1 = 8$ , $n = 30$ , and $a_{30} = 153$ into the sum formula:

$S_n = \frac{n}{2}(a_1 + a_n)$

$S_{30} = \frac{30}{2}(8 + 153)$

$S_{30} = 15(161)$

$S_{30} = 2{,}415$

The sum of the first 30 terms is $2{,}415$ . This example illustrates a two-step process commonly needed when applying the sum formula: first using the general term formula $a_n = a_1 + (n - 1)d$ to determine the last term, then substituting all known values into the sum formula $S_n = \frac{n}{2}(a_1 + a_n)$ .

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Updated 2026-05-25

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