/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Find each indicated sum. $$\su... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 23

Find each indicated sum. $$\sum_{i=1}^{4}\left(-\frac{1}{2}\right)^{i}$$

Short Answer

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Step by step solution

01

Identify the first term, and common ratio

The geometric series is defined as \(\sum_{i=1}^{4}\left(-\frac{1}{2}\right)^{i}\). The first term, \(a\) is \(-\frac{1}{2}\), obtained when \(i = 1\). The common ratio, \(r\), is also \(-\frac{1}{2}\). So, \(a = -\frac{1}{2}\) and \(r = -\frac{1}{2}\)
02

Compute the sum of geometric series

The formula for the sum \(S\) of a finite geometric series is \(S = a\frac{1-r^n}{1-r}\), where \(n\) is the number of terms. In this case, \(n = 4\). Substitute \(a = -\frac{1}{2}\), \(r = -\frac{1}{2}\), and \(n = 4\) into the formula to get \(S = -\frac{1}{2}\frac{1-(-\frac{1}{2})^4}{1 - -\frac{1}{2}}\).
03

Simplify the expression

Simplify the expression in Step 2 to get the final result. The term \((-1/2)^4\) simplifies to \(\frac{1}{16}\) and \(1 - \frac{1}{16}\) gives \(\frac{15}{16}\). The expression becomes \(S = -\frac{1}{2}\frac{\frac{15}{16}}{\frac{3}{2}}\), which simplifies to \(S = -\frac{5}{8}\)

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