How to Find the Sum of a Geometric Series

Problem

2 + 6 + 18 + 54 + 162

Answer

242

A geometric series multiplies by the same number at every step. When the number of terms is small you can add by hand, but the formula always works.

Step-by-step solution

Find the common ratio.
$r = \frac{6}{2} = 3$
Why: Each term is multiplied by the same number.
Count the terms.
$a = 2,\quad r = 3,\quad n = 5$
Why: The formula needs the first term, the ratio and the number of terms.
Write the sum formula.
$S = \frac{a(r^n - 1)}{r - 1}$
Why: The formula adds the first n terms of a geometric series.
Substitute and calculate.
$S = \frac{2(3^5 - 1)}{3 - 1} = \frac{2(242)}{2} = 242$
Why: 3 to the fifth power is 243, and 243 minus 1 is 242.

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