Practice Infinite Geometric Series in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The sum of all terms of a geometric sequence with common ratio ∣r∣<1|r| < 1. The infinite sum converges to a1−r\frac{a}{1-r}, where aa is the first term.

If each term is a fixed fraction of the previous one, the terms shrink fast enough that the total sum stays finite. Imagine walking halfway to a wall, then half the remaining distance, then half again—you approach the wall but the total distance is finite (exactly the full distance to the wall).

Showing a random 20 of 50 problems.

Example 1

challenge
The first term of a geometric series equals its common ratio: a=ra = r. If the sum is 13\frac{1}{3}, find rr.

Example 2

challenge
Find ∑n=1∞n2n\sum_{n=1}^{\infty} \frac{n}{2^n}.

Example 3

hard
An equilateral triangle has side 66. The midpoints form a smaller equilateral triangle, and this is repeated forever. Find the total perimeter of all triangles.

Example 4

medium
Find the sum of ∑n=2∞(13)n\sum_{n=2}^{\infty} \left(\frac{1}{3}\right)^n.

Example 5

medium
Express 0.45‾0.\overline{45} as a fraction using an infinite geometric series.

Example 6

easy
Find the sum of ∑n=0∞(27)n\sum_{n=0}^{\infty} \left(\frac{2}{7}\right)^n.

Example 7

easy
Find ∑n=0∞(13)n\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n.

Example 8

medium
A ball is dropped from 16 m and bounces to 12\frac{1}{2} its height each time. Find the total distance traveled.

Example 9

hard
For what values of xx does ∑n=0∞(x4)n\sum_{n=0}^{\infty} \left(\frac{x}{4}\right)^n converge? Give the sum on the interval of convergence.

Example 10

medium
A drug dose of 100 mg is taken daily; each day 20%20\% remains from prior doses. Find the long-run amount just after a dose.

Example 11

medium
If a geometric series has sum 92\frac{9}{2} and ratio r=13r = \frac{1}{3}, find the first term aa.

Example 12

hard
Find a closed form for ∑n=0∞(n+1)xn\sum_{n=0}^{\infty} (n+1) x^n when ∣x∣<1|x| < 1.

Example 13

medium
A geometric series has first term a=20a=20 and sum 8080. Find the common ratio rr.

Example 14

easy
Does ∑n=0∞(1.1)n\sum_{n=0}^{\infty} (1.1)^n converge or diverge?

Example 15

hard
Evaluate ∑n=0∞n+13n\sum_{n=0}^{\infty} \frac{n+1}{3^n}.

Example 16

easy
Find the sum of 1+12+14+18+⋯1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots.

Example 17

easy
Find ∑n=1∞(25)n\sum_{n=1}^{\infty} \left(\frac{2}{5}\right)^n.

Example 18

easy
For what values of rr does ∑n=0∞rn\sum_{n=0}^{\infty} r^n converge?

Example 19

medium
Find ∑n=0∞(−1)n2n\sum_{n=0}^{\infty} \frac{(-1)^n}{2^n}.

Example 20

medium
Express 0.16‾0.1\overline{6} as a fraction (the 6 repeats).
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