Determine whether the infinite geometric series has a finite sum. If so, find the limiting value. 112 + 56 + 28 + . . . A) S_∞ = 224 B) S_∞ = -224 C) S_∞ = 56 D) no finite sum

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Quizplus · Accepted Answer

The series has a finite sum because it is a geometric series with a common ratio $ r = \frac{1}{2} $ (each term is half the previous term), which is between -1 and 1. The sum of an infinite geometric series is given by $ S = \frac{a}{1 - r} $, where $ a $ is the first term and $ r $ is the common ratio. Here, $ a = 112 $ and $ r = \frac{1}{2} $, so $ S = \frac{112}{1 - \frac{1}{2}} = \frac{112}{\frac{1}{2}} = 224 $.

Determine Whether the Infinite Geometric Series Has a Finite Sum