/*! 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 11 Evaluate each geometric sum. $... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 11

Evaluate each geometric sum. $$\sum_{k=0}^{9}\left(-\frac{3}{4}\right)^{k}$$

Short Answer

Expert verified
Answer: The sum of the geometric series is approximately 2.082.

Step by step solution

01

Evaluate the first term '

' Set the value of \(k\) to 0 in the given expression to find the first term, \(a\): $$ a = \left(-\frac{3}{4}\right)^{0} = 1$$
02

Identify the common ratio '

' The common ratio, \(r\), is simply -3/4, which is the base of our expression: $$r = -\frac{3}{4}$$
03

Set the number of terms to sum '

' We have to sum from \(k=0\) to \(k=9\), so there are 10 terms to sum in total: $$n = 10$$
04

Apply the geometric series formula '

' Using the geometric series formula, we can compute the sum of this series: $$S_n = \frac{a(1 - r^n)}{1 - r}$$
05

Plug in the values and compute the sum '

' Substitute the values of \(a\), \(r\), and \(n\) into the geometric series formula, and compute the sum: $$S_{10} = \frac{1(1 - (-\frac{3}{4})^{10})}{1 - (-\frac{3}{4})}$$
06

Simplify the equation for the sum '

' Now, simplify the equation and solve for \(S_{10}\): $$S_{10} = \frac{1(1 - (-\frac{3}{4})^{10})}{\frac{7}{4}}$$ $$S_{10} = \frac{4(1 - (-\frac{3}{4})^{10})}{7}$$
07

Calculate the sum and obtain result '

' Perform the calculations to obtain the sum of the geometric series: $$S_{10} = \frac{4(1 - (-\frac{3}{4})^{10})}{7} \approx 2.082$$ So, the sum of the given geometric series is approximately 2.082.

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