Problem 38 - College Algebra

Question

Find the sum of each infinite geometric series. $$1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots$$

Step-by-Step Solution

Answer

\(\frac{4}{3}\)

Step 1: Identify the first term and the common ratio

The first term of the geometric series \(a\) is 1, and the common ratio \(r\) is obtained by dividing the second term by the first term, which gives \(\frac{1}{4}\).

Step 2: Apply the formula for the sum of an infinite geometric series

The formula for the sum \(S\) of an infinite geometric series is \(S = \frac{a}{1-r}\). Substituting the given values, \(S = \frac{1}{1-(\frac{1}{4})}\).

Step 3: Calculate the sum

This simplifies to \(S = \frac{1}{\frac{3}{4}}\), which further simplifies to \(S = \frac{4}{3}\). So the sum of the infinite geometric series is \(\frac{4}{3}\).