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Proof A sequence {a} is defined recursively by - a₁ = 1, a₂ = 3 and an = 2an-1 − An-2 for n ≥ 3. Then an = 2n – 1 for all n € N. - Wo progood by induction Singo the formulo holds for

Proof A sequence {a} is defined recursively by - a₁ = 1, a₂ = 3 and an = 2an-1 − An-2 for n ≥ 3. Then an = 2n – 1 for all n € N. - Wo progood by induction Singo the formulo holds for

Advanced Engineering Mathematics
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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Please explain what is being done step by step in the following proof, I understand induction but these proof puzzles me, i dont get why we take each step.

Result Proof A sequence {a} is defined recursively by a₁ = 1, a₂ = 3 and an = Then an 2n - 1 for all n E N. 2an-1-an-2 for n ≥ 3. We proceed by induction. Since a₁ = 2.1-1 = 1, the formula holds for n = 1. Assume for an arbitrary positive integer k that a; = 2i – 1 for all integers i with 1 ≤ i ≤ k. We show that ak+1 = 2(k + 1) − 1 = 2k + 1. If k = 1, then ak+1 = a₂ = 2·1+1 = 3. Since a2 = 3, it follows that ak+1 = 2k + 1 when k = 1. Hence, we may assume that k≥ 2. Since k + 1 ≥ 3, it follows that ak+1 2ak -ak-1 = 2(2k - 1) (2k-3) = 2k + 1, which is the desired result. By the Strong Principle of Mathematical Induction, an = 2n 1 for all n € N. ■
Transcribed Image Text:Result Proof A sequence {a} is defined recursively by a₁ = 1, a₂ = 3 and an = Then an 2n - 1 for all n E N. 2an-1-an-2 for n ≥ 3. We proceed by induction. Since a₁ = 2.1-1 = 1, the formula holds for n = 1. Assume for an arbitrary positive integer k that a; = 2i – 1 for all integers i with 1 ≤ i ≤ k. We show that ak+1 = 2(k + 1) − 1 = 2k + 1. If k = 1, then ak+1 = a₂ = 2·1+1 = 3. Since a2 = 3, it follows that ak+1 = 2k + 1 when k = 1. Hence, we may assume that k≥ 2. Since k + 1 ≥ 3, it follows that ak+1 2ak -ak-1 = 2(2k - 1) (2k-3) = 2k + 1, which is the desired result. By the Strong Principle of Mathematical Induction, an = 2n 1 for all n € N. ■
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