Use the formula for the sum of a geometric sequence to write the following sum in closed form. + 8k, where k is any integer with k ≥ 3.
Use the formula for the sum of a geometric sequence to write the following sum in closed form. + 8k, where k is any integer with k ≥ 3.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Use the formula for the sum of a geometric sequence to write the following sum in closed form.
\[ 8^3 + 8^4 + 8^5 + \ldots + 8^k \]
where \( k \) is any integer with \( k \geq 3 \).
---
**Explanation:**
To solve this problem, we need to use the formula for the sum of a geometric sequence. 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.
Here, the first term \( a = 8^3 \), the common ratio \( r = 8 \), and the number of terms is \( n = k - 3 + 1 = k - 2 \).
The formula for the sum of the first \( n \) terms of a geometric sequence is:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substitute the values:
\[ S = 8^3 \frac{8^{k-2} - 1}{8 - 1} \]
Simplify to find the closed form of the sum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2F59c813de-7783-471e-9eb4-c671da917f2b%2Fgiv67vh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the formula for the sum of a geometric sequence to write the following sum in closed form.
\[ 8^3 + 8^4 + 8^5 + \ldots + 8^k \]
where \( k \) is any integer with \( k \geq 3 \).
---
**Explanation:**
To solve this problem, we need to use the formula for the sum of a geometric sequence. 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.
Here, the first term \( a = 8^3 \), the common ratio \( r = 8 \), and the number of terms is \( n = k - 3 + 1 = k - 2 \).
The formula for the sum of the first \( n \) terms of a geometric sequence is:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substitute the values:
\[ S = 8^3 \frac{8^{k-2} - 1}{8 - 1} \]
Simplify to find the closed form of the sum.
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