Use the principle of mathematical induction to prove that 1 + r + r² +r³ + ... + µM =. pn+1 - r-1
Use the principle of mathematical induction to prove that 1 + r + r² +r³ + ... + µM =. pn+1 - r-1
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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![**Mathematical Induction on Geometric Series**
**Problem Statement:**
Use the principle of mathematical induction to prove that:
\[ 1 + r + r^2 + r^3 + \ldots + r^n = \frac{r^{n+1} - 1}{r - 1} \]
**Explanation:**
This problem involves proving a formula for the sum of a geometric series using mathematical induction. The left side of the equation represents a geometric series with first term 1 and common ratio \( r \), extending up to \( r^n \). The right side of the equation is the closed form expression of this series.
**Steps for Mathematical Induction:**
1. **Base Case:** Verify the formula for \( n = 1 \).
2. **Inductive Step:** Assume the formula holds for \( n = k \), i.e.,
\[ 1 + r + r^2 + \ldots + r^k = \frac{r^{k+1} - 1}{r - 1} \]
3. **Prove for \( n = k+1 \):** Show that
\[ 1 + r + r^2 + \ldots + r^k + r^{k+1} = \frac{r^{k+2} - 1}{r - 1} \]
By completing these steps, you show that the formula holds for all integers \( n \geq 1 \).
This mathematical proof technique is powerful for proving statements or formulas that are supposed to be true for all integers in a specified range.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7a1580d-b0b0-434b-847e-741d03f01d82%2Ffd3f48d4-2d8c-4538-bb65-f25a4f0996aa%2Fbzkhs8d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Induction on Geometric Series**
**Problem Statement:**
Use the principle of mathematical induction to prove that:
\[ 1 + r + r^2 + r^3 + \ldots + r^n = \frac{r^{n+1} - 1}{r - 1} \]
**Explanation:**
This problem involves proving a formula for the sum of a geometric series using mathematical induction. The left side of the equation represents a geometric series with first term 1 and common ratio \( r \), extending up to \( r^n \). The right side of the equation is the closed form expression of this series.
**Steps for Mathematical Induction:**
1. **Base Case:** Verify the formula for \( n = 1 \).
2. **Inductive Step:** Assume the formula holds for \( n = k \), i.e.,
\[ 1 + r + r^2 + \ldots + r^k = \frac{r^{k+1} - 1}{r - 1} \]
3. **Prove for \( n = k+1 \):** Show that
\[ 1 + r + r^2 + \ldots + r^k + r^{k+1} = \frac{r^{k+2} - 1}{r - 1} \]
By completing these steps, you show that the formula holds for all integers \( n \geq 1 \).
This mathematical proof technique is powerful for proving statements or formulas that are supposed to be true for all integers in a specified range.
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