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Chapter 7: Problem 22

Evaluate the geometric series. $$ 1+2+4+\cdots+2^{100} $$

Short Answer

Expert verified
The sum of the given geometric series is equal to \(2^{100} - 1\).

Step by step solution

01

Identify the first term, the common ratio, and the number of terms

In this geometric series, the first term is \(a_1 = 1\), the common ratio is \(r = 2\), and there are \(n = 100\) terms.
02

Plug the values into the geometric series sum formula

Use the formula for the sum of a geometric series: $$ S_n = \frac{a_1(1-r^n)}{1-r} $$ Substitute \(a_1 = 1, r = 2, n = 100\) into the formula: $$ S_{100} = \frac{1 (1 - 2^{100})}{1 - 2} $$
03

Simplify the expression

Now simplify the expression obtained in step 2: $$ S_{100} = \frac{1 (1 - 2^{100})}{-1} $$ Multiply the numerator and denominator by \(-1\) to eliminate the negative sign from the denominator: $$ S_{100} = \frac{1 (-1 + 2^{100})}{1} $$
04

Evaluate the sum of the geometric series

Now, we can remove the unnecessary denominator from the expression and calculate the final value for the sum of the geometric series: $$ S_{100} = - 1 + 2^{100} $$ The sum of the given geometric series is equal to \(2^{100} - 1\).

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