4. Use the PMI to prove the following for all natural numbers n. |(c) E-1 2' 2n+1 – 2 i=D1
4. Use the PMI to prove the following for all natural numbers n. |(c) E-1 2' 2n+1 – 2 i=D1
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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Sec 2.4 Pr 4
![**Problem 4: Use the Principle of Mathematical Induction (PMI) to prove the following statement for all natural numbers \( n \).**
**(c)** \( \sum_{i=1}^{n} 2^i = 2^{n+1} - 2 \)
In this problem, you are asked to use the Principle of Mathematical Induction to prove that the sum of the series \( \sum_{i=1}^{n} 2^i \) is equal to \( 2^{n+1} - 2 \) for all natural numbers \( n \).
*Steps Involved in Mathematical Induction:*
1. **Base Case**: Verify the statement for the initial value, typically \( n=1 \).
2. **Inductive Step**: Assume the statement holds for \( n = k \), and then prove it holds for \( n = k+1 \).
The given series and expression involve exponential growth, and confirming this identity through PMI solidifies your understanding of series summation and exponentiation properties in algebra.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9381f9ec-4907-481a-8789-2e44e2a64edb%2F15590b28-4f18-400e-952b-4a145832c8e9%2Fokx860c.png&w=3840&q=75)
Transcribed Image Text:**Problem 4: Use the Principle of Mathematical Induction (PMI) to prove the following statement for all natural numbers \( n \).**
**(c)** \( \sum_{i=1}^{n} 2^i = 2^{n+1} - 2 \)
In this problem, you are asked to use the Principle of Mathematical Induction to prove that the sum of the series \( \sum_{i=1}^{n} 2^i \) is equal to \( 2^{n+1} - 2 \) for all natural numbers \( n \).
*Steps Involved in Mathematical Induction:*
1. **Base Case**: Verify the statement for the initial value, typically \( n=1 \).
2. **Inductive Step**: Assume the statement holds for \( n = k \), and then prove it holds for \( n = k+1 \).
The given series and expression involve exponential growth, and confirming this identity through PMI solidifies your understanding of series summation and exponentiation properties in algebra.
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