Problem 90 Evaluate \(\sum_{n=1}^{5}(-1)^{n... [FREE SOLUTION]

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Precalculus Functions and Graphs

Mark Dugopolski

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 11: Problem 90

Evaluate $\sum_{n=1}^{5}(-1)^{n}(2 n-3)$

Short Answer

Expert verified

-3

Step by step solution

Understand the Summation Notation

The given expression is $\textstyle\big(危_{n=1}^{5}(-1)^{n}(2n-3)\textstyle\big)$. This means we need to evaluate the function $(-1)^{n}(2n-3)$ for each integer value of $n$ from 1 to 5 and then sum the results.

Evaluate the Function for Each Term

Calculate the value of the function $(-1)^{n}(2n-3)$ for $n=1,2,3,4,5$.When $n=1$: $(-1)^{1}(2\cdot1-3) = -1(-1) = 1$.When $n=2$: $(-1)^{2}(2\cdot2-3) = 1(4-3) = 1$.When $n=3$: $(-1)^{3}(2\cdot3-3) = -1(6-3) = -3$.When $n=4$: $(-1)^{4}(2\cdot4-3) = 1(8-3) = 5$.When $n=5$: $(-1)^{5}(2\cdot5-3) = -1(10-3) = -7$.

Sum the Evaluated Terms

Add the results from each term: $1 + 1 - 3 + 5 - 7$.First, combine the positive and negative terms separately: $1 + 1 + 5 = 7$ and $-3 - 7 = -10$.Then, sum these results: $7 + (-10) = -3$.

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Key Concepts

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

Summation Notation

Summation notation, often represented by the Greek letter 危, is a concise way of expressing the sum of a sequence of numbers. This notation eliminates the need to write repetitive addition operations.
In mathematics, summation is usually written as $ \sum_{i=a}^{b}f(i) $, where:

$i$ is the index of summation.
$a$ is the starting value of the index.
$b$ is the ending value of the index.
$f(i)$ is the function being summed.

For example, in $ \sum_{n=1}^{5}(-1)^{n}(2n-3)$, our index is \. The function we are evaluating is $ (-1)^n(2n-3) $ and the index runs from 1 to 5.

Alternating Series

An alternating series is a series where the terms alternate in sign. This means that one term is positive, the next term is negative, and so on. A typical alternating series has the form:
$ a_1 - a_2 + a_3 - a_4 + a_5 - \dots $ or $ (-1)^n a_n $, where the sign depends on the value of \. The given summation, $ \sum_{n=1}^{5}(-1)^{n}(2n-3) $, is an alternating series because $ (-1)^n $ changes sign for each term.
When evaluating alternating series, be careful to correctly apply the alternating sign to each term. This aspect can significantly impact the calculated result.

Evaluating Functions

Evaluating a function means calculating the output of a function for a particular input. In summation notation, each term in the sequence requires evaluating the function at specific integer values of the index. Here, we compute $ (-1)^n(2n-3) $ for \ n = 1 \ to \ n = 5 \.
Let's break it down step-by-step:

When \ n = 1 \: \ (-1)^{1}(2 \cdot 1 - 3) = -1(-1) = 1 \.
When \ n = 2 \: \ (-1)^{2}(2 \cdot 2 - 3) = 1(4 - 3) = 1 \.
When \ n = 3 \: \ (-1)^{3}(2 \cdot 3 - 3) = -1(6 - 3) = -3 \.
When \ n = 4 \: \ (-1)^{4}(2 \cdot 4 - 3) = 1(8 - 3) = 5 \.
When \ n = 5 \: \ (-1)^{5}(2 \cdot 5 - 3) = -1(10 - 3) = -7 \.

Finally, summing these evaluated terms \ ( 1 + 1 - 3 + 5 - 7 ) \ gives the result of $ -3 $.

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91影视

Evaluate \(\sum_{n=1}^{5}(-1)^{n}(2 n-3)\)

Short Answer

Step by step solution

Understand the Summation Notation

Evaluate the Function for Each Term

Sum the Evaluated Terms

Key Concepts

Summation Notation

Alternating Series

Evaluating Functions

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