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Chapter 8: Problem 14

Evaluate each geometric sum. $$\sum_{k=1}^{10}\left(\frac{4}{7}\right)^{k}$$

Short Answer

Expert verified
Answer: The sum of the first 10 terms of the given geometric series is $\frac{1126866692}{847425747}$.

Step by step solution

01

Identify the formula

We will use the formula for the sum of a geometric series: $$S_n = \frac{a_1(1 - r^n)}{1 - r}$$
02

Plug in the given values

We need to plug in the given values to the geometric sum formula. We have \(n = 10, a_1 = \left(\frac{4}{7}\right)^1 = \frac{4}{7}\), and \(r = \frac{4}{7}\). So the formula becomes: $$S_{10} = \frac{\left(\frac{4}{7}\right)(1 - \left(\frac{4}{7}\right)^{10})}{1 - \frac{4}{7}}$$
03

Calculate the numerical sum

Now, we will simply compute the numerical value of the sum. Start by computing the values of r^10 and then compute the denominator. Finally, compute the entire fraction: $$\left(\frac{4}{7}\right)^{10} = \frac{1048576}{282475249}$$ $$ 1 - \frac{4}{7} = \frac{3}{7} $$ Now we can plug these values back into the formula: $$S_{10} = \frac{\left(\frac{4}{7}\right)(1 - \frac{1048576}{282475249})}{\frac{3}{7}}$$
04

Simplify the expression

Simplify the expression by first computing the numerator and then divide it by the denominator: $$S_{10} = \frac{\left(\frac{4}{7}\right)\left(\frac{282475249 - 1048576}{282475249}\right)}{\frac{3}{7}}$$ $$S_{10} = \frac{4\left(\frac{282475249 - 1048576}{282475249}\right)}{3}$$ $$S_{10} = \frac{4(281716673)}{3(282475249)}$$ Therefore, the sum of the geometric series is: $$S_{10} = \frac{1126866692}{847425747}$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Sum
A geometric sum refers to the sum of terms in a geometric series. A geometric series is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. To understand a geometric sum, think of it like this: you are adding a series of terms that grow (or shrink) consistently by a common ratio.
  • Example: In the series \(2, 4, 8, 16\), the common ratio is 2, and the terms are multiplied by 2 to get each subsequent term.
  • The general expression for a geometric sum involves identifying the first term, common ratio, and number of terms in the sequence.
Understanding the concept of geometric sums is crucial, especially when dealing with lengthy arithmetic involving sequences that follow a pattern.
Sum Formula
The sum formula for a geometric series is a mathematical tool that allows you to find the sum of a series without adding each term individually. The formula is given by: \[ S_n = \frac{a_1(1 - r^n)}{1 - r} \]
  • \( S_n \) is the sum of the series up to the \( n \)-th term.
  • \( a_1 \) is the first term of the series.
  • \( r \) is the common ratio.
  • \( n \) is the number of terms in the series.
This formula is especially useful when dealing with a large number of terms. Mathematically, it negates the tedium of repeatedly multiplying and then adding up each term separately.
Series Evaluation
Evaluating a geometric series involves applying the sum formula to compute its total sum. This process typically involves a few clear steps:
  • First, determine the key attributes of the series, which are the first term \( a_1 \), the common ratio \( r \), and the number of terms \( n \).
  • Next, substitute these values into the sum formula to simplify for the numeric sum.
  • Finally, compute the result, carefully performing the arithmetic to ensure accuracy.
Evaluating a series is largely a combination of substitution and careful calculation. It requires understanding each component's role within the formula. In our example, we used \( r = \frac{4}{7} \), \( n = 10 \) to evaluate and arrive at the solution.
Step-by-Step Solution
A step-by-step solution breaks down the evaluation of a geometric series into manageable parts, making it easier to understand and follow. Here's how this is typically done:- **Identify the formula:** Recognize the sum formula to be used.- **Input known values:** Substitute the known values such as the first term \( a_1 \), the common ratio \( r \), and the number of terms \( n \) into the formula.- **Calculate:** Perform the mathematical operations to arrive at the sum.- **Simplify:** Streamline the expression as much as possible to get a clean final result.This step-by-step method ensures clarity and structure, assisting in reducing errors and increasing understanding. Such a methodical approach is invaluable in mathematics, where precision and accuracy are often paramount.

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Most popular questions from this chapter

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=n^{10} \text { and } b_{n}=n^{9} \ln ^{3} n, n \geq 7$$

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\(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} \operatorname{In} 1734,\) Leonhard Euler informally proved that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} .\) An elegant proof is outlined here that uses the inequality $$\cot ^{2} x<\frac{1}{x^{2}}<1+\cot ^{2} x\left(\text { provided that } 0

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