/*! 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 20 Express \(\sum_{i=0}^{5}(-2)^{i}... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 20

Express \(\sum_{i=0}^{5}(-2)^{i}\) without using summation notation.

Short Answer

Expert verified
-21

Step by step solution

01

- Identify the summation limits

The given summation notation is \(\textstyle \sum_{i=0}^{5}(-2)^{i}\). This means the summation starts at \(i = 0\) and ends at \(i = 5\).
02

- Expand the summation expression

We need to expand the summation from \(i = 0\) to \(i = 5\). This gives us the following series: \((-2)^{0} + (-2)^{1} + (-2)^{2} + (-2)^{3} + (-2)^{4} + (-2)^{5}\).
03

- Simplify the terms

Calculate each term of the series separately: \( (-2)^{0} = 1 \, \ (-2)^{1} = -2 \, \ (-2)^{2} = 4 \, \ (-2)^{3} = -8 \, \ (-2)^{4} = 16 \, \ (-2)^{5} = -32 \).
04

- Sum the simplified terms

Add the simplified terms together: \(1 + (-2) + 4 + (-8) + 16 + (-32)\).
05

- Calculate the final result

Perform the final addition: \(1 - 2 + 4 - 8 + 16 - 32 = -21\).

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Key Concepts

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

Series Expansion
The term 'series expansion' refers to writing out each term in a summation explicitly. This helps in understanding the elements involved. In our example, the summation notation \( \textstyle \sum_{i=0}^{5}(-2)^{i} \) can be expanded to:
\( \sum_{i=0}^{5}(-2)^{i} = (-2)^{0} + (-2)^{1} + (-2)^{2} + (-2)^{3} + (-2)^{4} + (-2)^{5} \). Breaking down summation notation in this way makes it easier to visualize each term in the series.
Initially, it might seem complex, but writing out the terms helps in performing the calculations step-by-step.
Simplification of Terms
Simplification is a key step in solving summation problems. After expanding the series, you need to simplify each term:
  • \((-2)^0 = 1\)
  • \((-2)^1 = -2\)
  • \((-2)^2 = 4\)
  • \((-2)^3 = -8\)
  • \((-2)^4 = 16\)
  • \((-2)^5 = -32\)

Calculating each term individually helps in accurately summing them later on. Simplification reduces the complexity by transforming the terms into more manageable numbers.
Final Summation Result
Once you have simplified each term, the final step is to add all the terms together. Here's how it looks:
\( 1 + (-2) + 4 + (-8) + 16 + (-32) \).
Performing the addition step-by-step:
  • \(1 - 2 = -1\)
  • \(-1 + 4 = 3\)
  • \(3 - 8 = -5\)
  • \(-5 + 16 = 11\)
  • \(11 - 32 = -21\)

So, the final summation result is -21. Summing the terms carefully ensures accuracy and helps in understanding the cumulative effect of each term in the series.

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