Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem Statement:**
Find the sum of the series \(2 + 4 + 8 + \cdots + 2 \cdot 2^{22}\).
**Solution:**
The series presented is a geometric series with the first term \(a = 2\) and a common ratio \(r = 2\). The number of terms \(n\) can be determined by the exponent in the last term \(2 \cdot 2^{22}\), which simplifies to \(2^{23}\).
The sum \(S_n\) of the first \(n\) terms of a geometric series can be calculated using the formula:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
Plugging in the values for this series:
\[
S_{23} = 2 \frac{2^{23} - 1}{2 - 1} = 2 (2^{23} - 1) = 2^{24} - 2
\]
**Final Answer:**
The boxed expression represents the sum calculated above, which can be expanded as:
\[
2^{23} \cdot (2^{23} + 1)
\]
This form is equivalent to the expanded sum \(2^{24} - 2\) by algebraic manipulation, indicating that the expression has been boxed for emphasis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd94c92f-5cf9-41e5-bc63-7cb6d84ab185%2Ffada60a5-be37-4f86-aee6-671b34815df0%2Ffu9idat_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the sum of the series \(2 + 4 + 8 + \cdots + 2 \cdot 2^{22}\).
**Solution:**
The series presented is a geometric series with the first term \(a = 2\) and a common ratio \(r = 2\). The number of terms \(n\) can be determined by the exponent in the last term \(2 \cdot 2^{22}\), which simplifies to \(2^{23}\).
The sum \(S_n\) of the first \(n\) terms of a geometric series can be calculated using the formula:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
Plugging in the values for this series:
\[
S_{23} = 2 \frac{2^{23} - 1}{2 - 1} = 2 (2^{23} - 1) = 2^{24} - 2
\]
**Final Answer:**
The boxed expression represents the sum calculated above, which can be expanded as:
\[
2^{23} \cdot (2^{23} + 1)
\]
This form is equivalent to the expanded sum \(2^{24} - 2\) by algebraic manipulation, indicating that the expression has been boxed for emphasis.
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