Question

one-a-day multi vitamins #18 5-4-3-2-1 5 multiple choice: do one-a-day! 1. the sum of the infinite geometric series \\(\frac{3}{2}+\frac{9}{16}+\frac{27}{128}+\frac{81}{1,024}+\dots\\) is (a) 1.60 (b) 2.35 (c) 2.40 (d) 2.45 (e) 2.50 no calculator

Explanation:

Step1: Identify the first term and common ratio

The first term \( a \) of the geometric series is \( \frac{3}{2} \). To find the common ratio \( r \), divide the second term by the first term: \( r=\frac{\frac{9}{16}}{\frac{3}{2}}=\frac{9}{16}\times\frac{2}{3}=\frac{3}{8} \).

Step2: Use the infinite geometric series sum formula

The formula for the sum \( S \) of an infinite geometric series is \( S = \frac{a}{1 - r} \) when \( |r| < 1 \). Substitute \( a=\frac{3}{2} \) and \( r=\frac{3}{8} \) into the formula:
\( S=\frac{\frac{3}{2}}{1-\frac{3}{8}}=\frac{\frac{3}{2}}{\frac{5}{8}}=\frac{3}{2}\times\frac{8}{5}=\frac{24}{10}=2.4 \)

Answer:

C. 2.40