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Sum of the First n Terms of an Arithmetic Sequence

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Example: Finding the Sum of an Arithmetic Sequence Given Its General Term

Find the sum of the first 50 terms of the arithmetic sequence whose general term is $a_n = 3n - 4$ .

To compute the sum, use the formula $S_n = \frac{n}{2}(a_1 + a_n)$ . The number of terms is $n = 50$ , but both $a_1$ and $a_{50}$ must be determined from the general term.

Find $a_1$ by substituting $n = 1$ :

$a_1 = 3(1) - 4 = -1$

Find $a_{50}$ by substituting $n = 50$ :

$a_{50} = 3(50) - 4 = 146$

Now substitute $n = 50$ , $a_1 = -1$ , and $a_{50} = 146$ into the sum formula:

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

$S_{50} = \frac{50}{2}(-1 + 146)$

$S_{50} = 25(145)$

$S_{50} = 3{,}625$

The sum of the first 50 terms is $3{,}625$ . When the general term formula is provided instead of specific sequence values, both the first and last terms are found by direct substitution into $a_n$ before applying the sum formula.

Updated 2026-05-25

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OpenStax Intermediate Algebra

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