Q2. Use induction to prove that for any natural number n , 2-1 = 2 – 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 2:**
Use induction to prove that for any natural number \( n \),
\[
\sum_{i=0}^{n} 2^{-i} = 2 - 2^{-n}.
\]
**Explanation:**
This question involves mathematical induction, a common proof technique used to establish the truth of an infinite sequence of statements. You are asked to use this method to prove a particular equality involving a geometric series. The equality states that the sum of powers of 2 with negative exponents from 0 to \( n \) equals \( 2 \) minus \( 2^{-n} \).
This typically involves two main steps:
1. **Base Case**: Verify that the equality holds for the initial value of \( n \), usually \( n = 0 \).
2. **Inductive Step**: Assume the statement holds for \( n = k \). Then, prove it also holds for \( n = k + 1 \).
Once both steps are accomplished, you will have shown that the equality is true for all natural numbers \( n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9696c01d-358b-4bf9-a3af-3184fd00a6a0%2F7586d814-410f-468e-977b-5453b69844d3%2Fmof76l_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 2:**
Use induction to prove that for any natural number \( n \),
\[
\sum_{i=0}^{n} 2^{-i} = 2 - 2^{-n}.
\]
**Explanation:**
This question involves mathematical induction, a common proof technique used to establish the truth of an infinite sequence of statements. You are asked to use this method to prove a particular equality involving a geometric series. The equality states that the sum of powers of 2 with negative exponents from 0 to \( n \) equals \( 2 \) minus \( 2^{-n} \).
This typically involves two main steps:
1. **Base Case**: Verify that the equality holds for the initial value of \( n \), usually \( n = 0 \).
2. **Inductive Step**: Assume the statement holds for \( n = k \). Then, prove it also holds for \( n = k + 1 \).
Once both steps are accomplished, you will have shown that the equality is true for all natural numbers \( n \).
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