How to Find the Sum of an Arithmetic Series

Problem

3 + 7 + 11 + \cdots \text{ (20 terms)}

Answer

S_{20} = 820

An arithmetic series adds the terms of an arithmetic sequence. The formula works for any number of terms once you know the first term and the difference.

Step-by-step solution

Read the first term and the common difference.
$a = 3,\quad d = 7 - 3 = 4$
Why: Each term adds 4, and the series starts at 3.
Write the sum formula for n terms.
$S_n = \frac{n}{2}\left(2a + (n - 1)d\right)$
Why: The formula pairs the first and last term and multiplies by the number of pairs.
Substitute a = 3, d = 4 and n = 20.
$S_{20} = \frac{20}{2}\left(2(3) + 19(4)\right)$
Why: There are 19 steps from the first term to the twentieth.
Simplify.
$S_{20} = 10(6 + 76) = 10 \cdot 82 = 820$
Why: The sum of the first 20 terms is 820.

Common mistakes

Using n instead of n - 1 for the number of steps.
Using the nth-term formula and reporting one term instead of the sum.
Multiplying 19 by 4 incorrectly. Write the arithmetic out when in doubt.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$1 + 4 + 7 + \cdots \text{ (10 terms)}$

Show answer

$S_{10} = 145$
$5 + 10 + 15 + \cdots \text{ (8 terms)}$

Show answer

$S_8 = 180$
$2 + 6 + 10 + \cdots \text{ (12 terms)}$

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$S_{12} = 288$
$10 + 8 + 6 + \cdots \text{ (10 terms)}$

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$S_{10} = 10$
$4 + 9 + 14 + \cdots \text{ (6 terms)}$

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$S_6 = 99$