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,...
Related questions
Question
100%
![**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.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images

Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you

Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON

Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning

Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning

Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON

Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning

Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning

Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON

Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press

College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education