Problem 42 - College Algebra

Question

Find the sum of each infinite geometric series. $$3-1+\frac{1}{3}-\frac{1}{9}+\cdots$$

Step-by-Step Solution

Answer

The sum of the given infinite geometric series is 2.25.

Step 1: Identify the first term

The first term of the series, denoted as \(a\), is the first number in the sequence. In this case, it is \(a = 3\).

Step 2: Find the common ratio

The common ratio (\(r\)) of a geometric series can be obtained by dividing any term in the series by its preceding term. In this case, divide the second term (-1) by the first term (3). So, \(r = -1 / 3 = -1/3\).

Step 3: Apply the sum formula of an infinite geometric series

The formula to find the sum \(S\) of an infinite geometric series is \(S = a / (1 - r)\). Substitute the values of \(a\) and \(r\) into the formula. This gives \(S = 3 / [1 - (-1/3)] = 3 / [1 + 1/3] = 3 / [4/3] = 9/4 = 2.25\).