Find the sum, if it exists, of the terms of the infinite geometric sequence.
-$\sum_{\mathrm{i}=1}^{\infty}(3)\left(-\frac{1}{4}\right)^{\mathrm{i}}$

A) $\frac{3}{5}$
B) $-\frac{3}{5}$
C) -1
D) Does not exist

Question

Find the sum, if it exists, of the terms of the infinite geometric sequence.
-$\sum_{\mathrm{i}=1}^{\infty}(3)\left(-\frac{1}{4}\right)^{\mathrm{i}}$

A) $\frac{3}{5}$
B) $-\frac{3}{5}$
C) -1
D) Does not exist

Quizplus · Accepted Answer

The sum of an infinite geometric series exists if the absolute value of the common ratio is less than 1. Here, the common ratio $ r = -\frac{1}{4} $, which satisfies this condition. The sum is given by $ S = \frac{a}{1-r} $, where $ a = 3 \times \left(-\frac{1}{4}\right) = -\frac{3}{4} $. Thus, $ S = \frac{-\frac{3}{4}}{1 - \left(-\frac{1}{4}\right)} = \frac{-\frac{3}{4}}{\frac{5}{4}} = -\frac{3}{5} $.

Find the Sum, If It Exists, of the Terms of the Infinite