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Chapter 11: Problem 44

Evaluate each series. $$\sum_{i=1}^{4}\left[(-2)^{i}-3\right]$$

Short Answer

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Step by step solution

01

Write down the series

The given series is \(\sum_{i=1}^{4}[(-2)^{i}-3]\). This means you need to evaluate the expression \((-2)^{i}-3\) for each integer value of \(i\) from 1 to 4 and then sum the results.
02

Evaluate the expression for each i

Calculate \((-2)^{i} - 3\) for each \(i\): When \(i=1\), \((-2)^{1} - 3 = -2 - 3 = -5\) When \(i=2\), \((-2)^{2} - 3 = 4 - 3 = 1\) When \(i=3\), \((-2)^{3} - 3 = -8 - 3 = -11\) When \(i=4\), \((-2)^{4} - 3 = 16 - 3 = 13\)
03

Add the results

Sum up the evaluated expressions: \(-5 + 1 + (-11) + 13\). Calculate the sum: \(-5 + 1 = -4\) \(-4 - 11 = -15\) \(-15 + 13 = -2\)

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Key Concepts

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

Understanding Sigma Notation
In mathematics, sigma notation is a compact way to write the sum of a sequence of terms. The Greek letter sigma, represented as \( \sum \), is used to indicate summation. Typically, sigma notation looks like this: \( \sum_{i=a}^{b} f(i) \), where:
  • \( i \) is the index of summation, starting at \( a \) and ending at \( b \).
  • \( f(i) \) is the function or expression involving the index \( i \).
In the given exercise, the series is written as \( \sum_{i=1}^{4}[(-2)^{i}-3] \). Here, the function \( f(i) = (-2)^{i}-3 \) is evaluated from \( i = 1 \) to \( i = 4 \). Understanding how to manipulate and compute the terms in sigma notation is essential to accurately finding the sum of a series.
Arithmetic Sequence Basics
An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant difference, known as the common difference, to the preceding term. The sequence follows the form: \( a, a+d, a+2d, ... \), where:
  • \( a \) denotes the first term.
  • \( d \) is the common difference between terms.
Although the series in this exercise uses a different kind of sequence, arithmetic sequences are a foundational concept in understanding more complex series and summations. Recognizing patterns in terms is crucial for evaluating them correctly. The practice of breaking down each step, as shown in the exercise solution, aligns with methods used for arithmetic sequences to make sure calculations are accurate.
Sum of a Series (Summation)
Summation is the process of adding a sequence of numbers, typically expressed in a compact form using sigma notation. It involves calculating the aggregate total of terms within a specific range. In the given problem, summation is used to find the total of the series values from \( i = 1 \) to \( i = 4 \). Let's break down the steps clearly:
  • First, evaluate each term of the function \( (-2)^{i} - 3 \) for every value of \( i \).
  • Next, list all resulting terms: \( -5, 1, -11, 13 \).
  • Finally, add these values together sequentially: \(-5 + 1 = -4 \), \( -4 - 11 = -15 \), and \(-15 + 13 = -2 \).
The resulting summation gives a total of \( -2 \). Understanding how to systematically approach and execute these steps is crucial for solving more complex summation problems in higher mathematics.

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Most popular questions from this chapter

Use the sequence feature of a graphing calculator to graph the first ten terms of each sequence as defined. Use the graph to make a conjecture as to whether the sequence converges or diverges. If you think it converges, determine the number to which it converges. $$a_{n}=(1+n)^{1 / n}$$

A financial analyst has determined the possibilities (and their probabilities) for the growth in value of a certain stock during the next year. (Assume these are the only possibilities.) See the table. For instance, the probability of a \(5 \%\) growth is \(0.15 .\) If you invest \(\$ 10,000\) in the stock, what is the probability that the stock will be worth at least \(\$ 11,400\) by the end of the year? $$\begin{array}{c|c}\hline \text { Percent Growth } & \text { Probability } \\\\\hline 5 & 0.15 \\\\\hline 8 & 0.20 \\\\\hline 10 & 0.35 \\\\\hline 14 & 0.20 \\\\\hline 18 & 0.10 \\\\\hline\end{array}$$

Five cards are marked with the numbers \(1,2,3,4,\) and \(5,\) shuffled, and 2 cards are then drawn. How many different 2 -card hands are possible?

In an experiment on plant hardiness, a researcher gathers 6 wheat plants, 3 barley plants, and 2 rye plants. She wishes to select 4 plants at random. (a) In how many ways can this be done? (b) In how many ways can this be done if exactly 2 wheat plants must be included?

Use the summation properties and rules to evaluate each series. $$\sum_{i=1}^{4}\left(3 i^{3}+2 i-4\right)$$
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