### Problem DescriptionFind the sum; write your answer as a fraction\[ \sum_{k=1}^{8} 3 \left(\frac{1}{4}\right)^{k-1} \]### ExplanationThis is a summation problem where you need to find the sum of a geometric series. The notation \(\sum_{k=1}^{8}\) indicates that we are summing from \(k=1\) to \(k=8\). Each term in the series is given by the formula \(3 \left(\frac{1}{4}\right)^{k-1}\).### Solution Steps1. **Identify the first term and common ratio:** - First term (\(a\)): When \(k=1\), the term is \(3 \left(\frac{1}{4}\right)^{1-1} = 3(1) = 3\). - Common ratio (\(r\)): The base of the exponent in the term formula is \(\frac{1}{4}\).2. **Use the formula for the sum of first \(n\) terms of a geometric series:** \[ S_n = \frac{a(1 - r^n)}{1 - r} \] Here, \(a = 3\), \(r = \frac{1}{4}\), and \(n = 8\).3. **Substitute the values into the formula:** \[ S_8 = \frac{3(1 - (\frac{1}{4})^8)}{1 - \frac{1}{4}} \]4. **Calculate the power and difference:** \[ (\frac{1}{4})^8 = \frac{1}{65536} \] \[ 1 - \frac{1}{65536} = \frac{65536 - 1}{65536} = \frac{65535}{65536} \]5. **Substitute back into the sum formula:** \[ S_8 = \frac{3 \cdot \frac{65535}{65536}}{\frac{3}{4}} = \frac{3 \cdot 65535}{65536} \times \frac{4}{3} \] \[ S_8

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Answered: Find the sum; write your answer as a…

Find the sum; write your answer as a fraction A-1 3. kw1

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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