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Chapter 11: Problem 39

Find and evaluate the sum. $$\sum_{i=0}^{6} 2^{i}$$

Short Answer

Expert verified
The sum is 127.

Step by step solution

01

Recognize the Summation

Identify that the exercise is to find the sum of a series which can be expressed as \(\sum_{i=0}^{6} 2^{i}\). This means adding the terms where the exponent of 2 changes from 0 to 6.
02

List the Terms

Write out the individual terms: \(2^{0}, 2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5}, 2^{6}\).
03

Evaluate Each Term

Calculate the value of each term: \(2^{0} = 1, 2^{1} = 2, 2^{2} = 4, 2^{3} = 8, 2^{4} = 16, 2^{5} = 32, 2^{6} = 64\).
04

Sum the Terms

Add the evaluated terms together: \(1 + 2 + 4 + 8 + 16 + 32 + 64\).
05

Calculate the Total Sum

Perform the addition to find the total: \(1 + 2 + 4 + 8 + 16 + 32 + 64 = 127\).

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Key Concepts

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

Summation Notation
Summation notation, denoted by the Greek letter sigma \(\sum\), is a powerful mathematical tool used to concisely express the sum of a sequence of terms. In our exercise, we have \(\sum_{i=0}^{6} 2^{i}\). This denotes the sum of terms where the exponent of 2 starts at zero and ends at six.
The notation is broken down as follows:
  • \(i=0\): The starting index of the summation.
  • \(6\): The ending index of the summation.
  • \(2^{i}\): The general term to be summed, in this case, it is an exponential function of the form \(2^{i}\).
Summation notation simplifies the representation of adding long sequences. It is especially useful for communicating complex mathematical concepts in a more readable form.
Exponential Functions
An exponential function involves a base raised to an exponent. In our exercise, the base is 2 and the exponent varies from 0 to 6. Let's refresh the concept of exponential functions:
  • Base (here 2): The number that gets multiplied.
  • Exponent (here \(i\): The number of times the base is multiplied by itself.
Computing these, we get:
  • \(2^{0} = 1\)
  • \(2^{1} = 2\)
  • \(2^{2} = 4\)
  • \(2^{3} = 8\)
  • \(2^{4} = 16\)
  • \(2^{5} = 32\)
  • \(2^{6} = 64\)
An exponential function grows very quickly due to the nature of repeated multiplication. This is why our individual terms increase rapidly as the exponent increases.
Series Evaluation
Evaluating a series involves adding together the terms defined by the summation notation. Here are the steps broken down:
First, we write the terms using the formula \(2^{i}\) for \(i\) from 0 to 6:
  • \(2^{0}=1\)
  • \(2^{1}=2\)
  • \(2^{2}=4\)
  • \(2^{3}=8\)
  • \(2^{4}=16\)
  • \(2^{5}=32\)
  • \(2^{6}=64\)
Next, we add these terms together to get the total sum:
\(1 + 2 + 4 + 8 + 16 + 32 + 64 = 127\)
This sum, 127, is the result of evaluating the series from \(i=0\) to \(i=6\) for the function \(2^{i}\). Breaking down the process step by step makes it manageable and prevents any confusion.

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