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
I need help with this question so badly help please give the right answer
![**Problem Statement:**
Find the sum of the first 10 terms of the following series, to the nearest integer.
**Given Series:**
10, 40, 160, ...
---
To solve this, we need to identify the type of series and then apply the appropriate formula. The given series 10, 40, 160, ... is a geometric sequence because each term after the first is obtained by multiplying the previous term by a constant ratio.
**Step-by-Step Solution:**
1. **Identify the First Term (a):**
- The first term \( a \) is 10.
2. **Calculate the Common Ratio (r):**
- The common ratio \( r \) is obtained by dividing the second term by the first term, or the third term by the second term.
- \( r = \frac{40}{10} = 4 \)
- This can also be verified by \( r = \frac{160}{40} = 4 \).
3. **Sum of the First n Terms (S_n) of a Geometric Sequence:**
- The formula for the sum of the first \( n \) terms of a geometric sequence is:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
- Substituting the known values into the formula to find the sum of the first 10 terms:
\[
S_{10} = 10 \frac{4^{10} - 1}{4 - 1}
\]
4. **Simplify the Expression:**
- Calculate \( 4^{10} \):
\[
4^{10} = 1048576
\]
- Substitute \( 1048576 \) into the sum formula:
\[
S_{10} = 10 \frac{1048576 - 1}{3} = 10 \frac{1048575}{3} = 10 \cdot 349525 = 3495250
\]
5. **Round to the Nearest Integer:**
- The sum of the first 10 terms rounded to the nearest integer is \( \boxed{3495250} \).
---
**Conclusion:**
The sum of the first 10 terms of the given geometric series 10, 40, 160, ... is approximately 3,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa039d78a-0316-4741-8eaf-c3b23c002a7a%2Fcc3e501c-3916-4a39-8ddb-13e5fbee5d72%2Fmbb22z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the sum of the first 10 terms of the following series, to the nearest integer.
**Given Series:**
10, 40, 160, ...
---
To solve this, we need to identify the type of series and then apply the appropriate formula. The given series 10, 40, 160, ... is a geometric sequence because each term after the first is obtained by multiplying the previous term by a constant ratio.
**Step-by-Step Solution:**
1. **Identify the First Term (a):**
- The first term \( a \) is 10.
2. **Calculate the Common Ratio (r):**
- The common ratio \( r \) is obtained by dividing the second term by the first term, or the third term by the second term.
- \( r = \frac{40}{10} = 4 \)
- This can also be verified by \( r = \frac{160}{40} = 4 \).
3. **Sum of the First n Terms (S_n) of a Geometric Sequence:**
- The formula for the sum of the first \( n \) terms of a geometric sequence is:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
- Substituting the known values into the formula to find the sum of the first 10 terms:
\[
S_{10} = 10 \frac{4^{10} - 1}{4 - 1}
\]
4. **Simplify the Expression:**
- Calculate \( 4^{10} \):
\[
4^{10} = 1048576
\]
- Substitute \( 1048576 \) into the sum formula:
\[
S_{10} = 10 \frac{1048576 - 1}{3} = 10 \frac{1048575}{3} = 10 \cdot 349525 = 3495250
\]
5. **Round to the Nearest Integer:**
- The sum of the first 10 terms rounded to the nearest integer is \( \boxed{3495250} \).
---
**Conclusion:**
The sum of the first 10 terms of the given geometric series 10, 40, 160, ... is approximately 3,
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 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