Infinite Geometric Series and Convergence Practice

Determine whether the series 12 + 6 + 3 + ... converges. If so, find its sum.

Converges; r = 1/2. S = 12 / (1 - 1/2) = 24.

Determine whether the series 5 - 10 + 20 - 40 + ... converges.

Diverges; r = -2, and |-2| >= 1.

Find the sum of the series 27 - 9 + 3 - 1 + ...

Converges; a = 27, r = -1/3. S = 27 / (1 - (-1/3)) = 27 / (4/3) = 81/4 (or 20.25).

Write the repeating decimal 0.777... as an infinite geometric series and calculate its fraction form.

0.7 + 0.07 + 0.007 + ... with a = 0.7 and r = 0.1. S = 0.7 / (1 - 0.1) = 7/9.

An infinite geometric series has a sum of 40 and a common ratio of 3/4. Find its first term a.

a = 10. From S = a / (1 - r), 40 = a / (1 - 3/4) = a / (1/4), so a = 40 * 1/4 = 10.

The first term of an infinite geometric series is 15 and its sum is 20. Find the common ratio r.

r = 1/4. From 20 = 15 / (1 - r), 1 - r = 15/20 = 3/4, so r = 1 - 3/4 = 1/4.