Consider the sequence {an}, defined by ai = 2 and an =6(2an-1 + 5) for n > 2.n=1Use induction to prove that an >1 for all n E J.

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Description: Consider the sequence {an}, defined by ai = 2 and an =6(2an-1 + 5) for n > 2.n=1Use induction to prove that an >1 for all n E J.

Given that a1=2 and an=16(2an-1+5) The objective is to show that an≥1. Let's prove the following…

Answered: Consider the sequence {an}, defined by…

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Advanced Math

Consider the sequence {an}, defined by a1 = 2 and an = E (2an-1 + 5) for n > 2. Use induction to prove that an 21 for all n E J.

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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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

Consider the sequence {an}, defined by ai = 2 and an =
6(2an-1 + 5) for n > 2.
n=1
Use induction to prove that an >1 for all n E J.

Expert Solution

Step 1

Given that $a_{1} = 2$ and $a_{n} = \frac{1}{6} (2 a_{n - 1} + 5)$

The objective is to show that $a_{n} \geq 1$ .

Let's prove the following using induction.

For $n = 1$

$a_{1} = 2 > 1$

Hence $a_{n} \geq 1$ for n=1.

Now let's assume that for $n = k$ , $a_{k} \geq 1$ is true.

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