Consider the sequence {an}=1 defined by aj = 2 and an =(2an-1 + 5) for n > 2.Use induction to prove that {an}1 is decreasing.n=1

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Description: Consider the sequence {an}=1 defined by aj = 2 and an =(2an-1 + 5) for n > 2.Use induction to prove that {an}1 is decreasing.n=1

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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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Consider the sequence {an}=1 defined by aj = 2 and an =
(2an-1 + 5) for n > 2.
Use induction to prove that {an}1 is decreasing.
n=1

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Introduction

Proof by induction for a given statement deals with first proving the statement is true for $n = 1$ and assuming the statement is also true for $n = k$ and finally showing the statement is true for $n = k + 1$ . then only the given statement is said to be proved for its truth.

For any sequence ${\{a\}}_{n = 1}^{\infty}$ to prove that it is decreasing it is enough to prove that for every n $a_{n} < a_{n - 1}$ is true, then only the sequence ${\{a\}}_{n = 1}^{\infty}$ is said to be decreasing.

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