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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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,
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