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Chapter 8: Problem 12

Evaluate each geometric sum. $$\sum_{k=1}^{5}(-2.5)^{k}$$

Short Answer

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Question: Find the sum of the given geometric series: $$\sum_{k=1}^{5}(-2.5)^{k}$$ Answer: The sum of the given geometric series is approximately -69.6875.

Step by step solution

01

Identify the necessary components of the geometric series

The given sum is $$\sum_{k=1}^{5}(-2.5)^{k}$$ In this geometric series, the first term (\(a_1\)) is \((-2.5)^{1} = -2.5\), the common ratio (r) is -2.5, and the number of terms (n) is 5.
02

Use the formula for the sum of a finite geometric series

The formula for the sum of a finite geometric series is: $$S = \frac{a_1(1-r^n)}{1-r}$$. Plug in the values for \(a_1\), r, and n: $$S = \frac{-2.5\big(1 - (-2.5)^5\big)}{1 - (-2.5)}$$
03

Simplify the expression

First, calculate \((-2.5)^5 = -97.65625\). Then, plug this into the expression: $$S = \frac{-2.5(1 - (-97.65625))}{1 - (-2.5)}$$ Now, simplify the expression in the numerator: $$1 - (-97.65625) = 98.65625$$ Multiply this by -2.5: $$-2.5\cdot 98.65625 = -243.90625$$ Now, simplify the expression in the denominator: $$1 - (-2.5) = 3.5$$
04

Calculate the final result

Divide the numerator (-243.90625) by the denominator (3.5) to get the final sum: $$S = \frac{-243.90625}{3.5} \approx -69.6875$$ So, the geometric sum is approximately -69.6875.

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Key Concepts

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

Finite Geometric Series
A finite geometric series is a sum of a sequence of numbers where each number after the first is multiplied by a fixed, non-zero number known as the common ratio. Specifically, a finite geometric series consists of a finite number of terms, which makes it possible to calculate an exact sum.

Understanding how a finite geometric series works is crucial because it appears in various mathematical contexts, from simple interest calculations in finance to the behavior of certain physical systems. When approaching these series, it's essential to identify the first term, the common ratio, and the total number of terms to proceed with calculations.
Geometric Series Formula
To calculate the sum of a finite geometric series, a specific formula is used: \[ S = \frac{a_1(1-r^n)}{1-r} \]. Here, \( S \) is the sum of the series, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms in the series.

This formula is a powerful tool as it simplifies the process of finding the sum without having to manually add each term. To properly use this formula, it鈥檚 important to precisely determine the components of the series and then execute each operation in the formula with care to avoid any calculation errors.
Simplifying Expressions
Simplifying expressions is a fundamental skill in mathematics that involves reducing an expression to its simplest form. This process makes calculations easier and helps to understand the structure of the expression. To simplify expressions in the context of a geometric series sum, one must carry out exponentiation, handle negative numbers, and simplify fractions.

For example, exponentiating can turn numbers into large or small values, and it is common to deal with subtraction of negative numbers, which effectively changes the operation to addition. Furthermore, simplifying fractions involves dividing the numerator by the denominator to find the simplified sum of the series.
Exponentiation
Exponentiation is a mathematical operation that involves raising a base number to the power of an exponent, indicating how many times the base is multiplied by itself. In the context of geometric series, exponentiation often results in very large or small numbers, significantly affecting the sum.

Understanding exponentiation is pivotal when working with geometric series. If an exponent is positive, the base will increase in magnitude; if it is negative, the result will be a fraction. Additionally, this operation can involve non-integer bases, as seen with the common ratio in the given geometric series. Correctly performing exponentiation ensures accurate calculation of the series sum.

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Most popular questions from this chapter

A fallacy Explain the fallacy in the following argument. Let \(x=1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots\) and \(y=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\cdots \cdot\) It follows that \(2 y=x+y\) which implies that \(x=y .\) On the other hand, $$ x-y=\underbrace{\left(1-\frac{1}{2}\right)}_{>0}+\underbrace{\left(\frac{1}{3}-\frac{1}{4}\right)}_{>0}+\underbrace{\left(\frac{1}{5}-\frac{1}{6}\right)}_{>0}+\cdots>0 $$ is a sum of positive terms, so \(x>y .\) Therefore, we have shown that \(x=y\) and \(x>y\)

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. Population growth When a biologist begins a study, a colony of prairie dogs has a population of \(250 .\) Regular measurements reveal that each month the prairie dog population increases by \(3 \%\) Let \(p_{n}\) be the population (rounded to whole numbers) at the end of the \(n\) th month, where the initial population is \(p_{0}=250\).

Consider series \(S=\sum_{k=0}^{n} r^{k},\) where \(|r|<1\) and its sequence of partial sums \(S_{n}=\sum_{k=0}^{n} r^{k}\) a. Complete the following table showing the smallest value of \(n,\) calling it \(N(r),\) such that \(\left|S-S_{n}\right|<10^{-4},\) for various values of \(r .\) For example, with \(r=0.5\) and \(S=2,\) we find that \(\left|S-S_{13}\right|=1.2 \times 10^{-4}\) and \(\left|S-S_{14}\right|=6.1 \times 10^{-5}\) Therefore, \(N(0.5)=14\) $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline r & -0.9 & -0.7 & -0.5 & -0.2 & 0 & 0.2 & 0.5 & 0.7 & 0.9 \\ \hline N(r) & & & & & & & 14 & & \\ \hline \end{array}$$ b. Make a graph of \(N(r)\) for the values of \(r\) in part (a). c. How does the rate of convergence of the geometric series depend on \(r ?\)

James begins a savings plan in which he deposits \(\$ 100\) at the beginning of each month into an account that earns \(9 \%\) interest annually or, equivalently, \(0.75 \%\) per month. To be clear, on the first day of each month, the bank adds \(0.75 \%\) of the current balance as interest, and then James deposits \(\$ 100\). Let \(B_{n}\) be the balance in the account after the \(n\) th deposit, where \(B_{0}=\$ 0\). a. Write the first five terms of the sequence \(\left\\{B_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{B_{n}\right\\}\). c. How many months are needed to reach a balance of \(\$ 5000 ?\)

Consider the geometric series \(S=\sum_{k=0}^{\infty} r^{k}\) which has the value \(1 /(1-r)\) provided \(|r|<1\). Let \(S_{n}=\sum_{k=0}^{n-1} r^{k}=\frac{1-r^{n}}{1-r}\) be the sum of the first \(n\) terms. The magnitude of the remainder \(R_{n}\) is the error in approximating \(S\) by \(S_{n} .\) Show that $$ R_{n}=S-S_{n}=\frac{r^{n}}{1-r} $$
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