Use the principle of mathematical induction to complete the proof that 3+3² +33 +...+ 3" = 3+1 -3 2
Use the principle of mathematical induction to complete the proof that 3+3² +33 +...+ 3" = 3+1 -3 2
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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Question
![**Mathematical Induction Proof**
**Problem:**
Use the principle of mathematical induction to complete the proof that \(3 + 3^2 + 3^3 + \ldots + 3^n = \frac{3^{n+1} - 3}{2}\).
---
**Let \(P(n)\) be the proposition:**
__________________________________________________
**Step 1:** Verify that \(P(1)\) is true.
__________________________________________________
**Step 2:** Assume that ____________________ is true. That is, assume that
__________________________________________________
**Step 3:** Prove that ____________________ is true. That is, prove that
(1) __________________________________________________ Induction hypothesis
Thus,
(2) \(3 + 3^2 + 3^3 + \ldots + 3^k + 3^{k+1} =\)
\[
= \frac{3^{k+2} - 3}{2}
\]
---
The proof uses the principle of mathematical induction, which involves three key steps: verifying the base case \(P(1)\), assuming the statement \(P(k)\) is true for some \(k\), and proving that \(P(k+1)\) is true based on that assumption.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7a1580d-b0b0-434b-847e-741d03f01d82%2F99deacd2-c4ac-4070-8e1d-e6a31fea16ce%2Fqaepeps_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Induction Proof**
**Problem:**
Use the principle of mathematical induction to complete the proof that \(3 + 3^2 + 3^3 + \ldots + 3^n = \frac{3^{n+1} - 3}{2}\).
---
**Let \(P(n)\) be the proposition:**
__________________________________________________
**Step 1:** Verify that \(P(1)\) is true.
__________________________________________________
**Step 2:** Assume that ____________________ is true. That is, assume that
__________________________________________________
**Step 3:** Prove that ____________________ is true. That is, prove that
(1) __________________________________________________ Induction hypothesis
Thus,
(2) \(3 + 3^2 + 3^3 + \ldots + 3^k + 3^{k+1} =\)
\[
= \frac{3^{k+2} - 3}{2}
\]
---
The proof uses the principle of mathematical induction, which involves three key steps: verifying the base case \(P(1)\), assuming the statement \(P(k)\) is true for some \(k\), and proving that \(P(k+1)\) is true based on that assumption.
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