Problem 44 - College Algebra

Question

Find the sum of each infinite geometric series. $$\sum_{i=1}^{\infty} 12(-0.7)^{i-1}$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is approximately 7.06.

Step 1: Identify the sequence and ratio

First, note that the given series is a geometric series because each term is multiplied by a constant to arrive at the next. In this case, our first term \(a = 12\) and our common ratio \(r = -0.7\).

Step 2: Verify the conditions

In order to find the sum of an infinite geometric series, the series must be converging. For the series to converge, the absolute value of the common ratio \(|r|\) must be less than 1. So checking the condition here, \(|-0.7|< 1\); the series converges.

Step 3: Use the geometric series sum formula

For a geometric series with first term \(a\) and common ratio \(r\), the sum (\(S\)) of the series can be found using the formula \(S = \frac{a}{{1 - r}}\). Here, \(S = \frac{12}{{1 - (-0.7)}}\).

Step 4: Solve the calculation

Completing the calculation gives us \(S = \frac{12}{{1.7}} = 7.06\) (rounded to 2 decimal places). Therefore, the sum of the given infinite series is approximately 7.06.