The sum of the first 10 terms of a geometric sequence is equal to 1456/243. (a) Find the first term, u₁, if r = %. (b) Find the sum to infinity of this sequence.
The sum of the first 10 terms of a geometric sequence is equal to 1456/243. (a) Find the first term, u₁, if r = %. (b) Find the sum to infinity of this sequence.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Geometric Sequences**
The sum of the first 10 terms of a geometric sequence is equal to \( \frac{1456}{243} \).
(a) **Find the first term, \( u_1 \), if \( r = \frac{1}{3} \).**
(b) **Find the sum to infinity of this sequence.**
---
### Explanation:
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (\( r \)). The general formula for the sum of the first \( n \) terms of a geometric sequence is:
\[ S_n = u_1 \frac{1-r^n}{1-r} \]
For the sum to infinity, the formula is:
\[ S_{\infty} = \frac{u_1}{1-r} \]
provided that \( |r| < 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d6cf028-a950-4acd-9ac8-174a6628eca5%2F47f1638e-efa4-406b-a9ed-69ef8f3b2f00%2Fs9w82v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Geometric Sequences**
The sum of the first 10 terms of a geometric sequence is equal to \( \frac{1456}{243} \).
(a) **Find the first term, \( u_1 \), if \( r = \frac{1}{3} \).**
(b) **Find the sum to infinity of this sequence.**
---
### Explanation:
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (\( r \)). The general formula for the sum of the first \( n \) terms of a geometric sequence is:
\[ S_n = u_1 \frac{1-r^n}{1-r} \]
For the sum to infinity, the formula is:
\[ S_{\infty} = \frac{u_1}{1-r} \]
provided that \( |r| < 1 \).
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