Let a, be the nth term of the sequence defined recursively by1an + 11 + anand a, = 1. Find a formula for a, in terms of the Fibonacci numbers Fp.an =Prove that the formula you found is valid for all natural numbers n.Let P(n) denote the statement that a, == 1, which is true.P(1) is the statement that a, =Assume that P(k) is true. Thus, our induction hypothesis is ak =We want to use this to show that P(k + 1) is true. Now,1ak+1 =1 + ak11 +Thus, P(k + 1) follows from P(k), and this completes the induction step. Having proven the above steps, we conclude by the Principle of Mathematical Induction that P(n) is truefor all n 2 1.

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Description: Let a, be the nth term of the sequence defined recursively by1an + 11 + anand a, = 1. Find a formula for a, in terms of the Fibonacci numbers Fp.an =Prove that the formula you found is valid for all natural numbers n.Let P(n) denote the statement th

Using recursive relation, we get a1=1=11; a2=11+a1=12; a3=11+a2=11+12=132=23; a4=11+a3=11+23=153=35;…

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Let a, be the nth term of the sequence defined recursively by an + 1 1 + a and a, = 1. Find a formula for a, in terms of the Fibonacci numbers F. an = Prove that the formula you found is valid for all natural numbers n. Let P(n) denote the statement that a, = P(1) is the statement that a, = = 1, which is true. Assume that P(k) is true. Thus, our induction hypothesis is ak = We want to use this to show that P(k + 1) is true. Now, ak+1 = 1+ ak %3! 1 + Thus, P(k + 1) follows from P(k), and this completes the induction step. Having proven the above steps, we conclude by the Principle of Mathematical Induction that P(n) is true for all n 2 1.

Let a, be the nth term of the sequence defined recursively by an + 1 1 + a and a, = 1. Find a formula for a, in terms of the Fibonacci numbers F. an = Prove that the formula you found is valid for all natural numbers n. Let P(n) denote the statement that a, = P(1) is the statement that a, = = 1, which is true. Assume that P(k) is true. Thus, our induction hypothesis is ak = We want to use this to show that P(k + 1) is true. Now, ak+1 = 1+ ak %3! 1 + Thus, P(k + 1) follows from P(k), and this completes the induction step. Having proven the above steps, we conclude by the Principle of Mathematical Induction that P(n) is true for all n 2 1.

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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Let a, be the nth term of the sequence defined recursively by
1
an + 1
1 + an
and a, = 1. Find a formula for a, in terms of the Fibonacci numbers Fp.
an =
Prove that the formula you found is valid for all natural numbers n.
Let P(n) denote the statement that a, =
= 1, which is true.
P(1) is the statement that a, =
Assume that P(k) is true. Thus, our induction hypothesis is ak =
We want to use this to show that P(k + 1) is true. Now,
1
ak+1 =
1 + ak
1
1 +
Thus, P(k + 1) follows from P(k), and this completes the induction step. Having proven the above steps, we conclude by the Principle of Mathematical Induction that P(n) is true
for all n 2 1.

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