Analysis Consider the sequence defined by tn = ff, where f; is the jth term in the Fibonaccisequence (recall that f₁ = f2 = 1 and fn+1 = fn + fn-1).(a) Prove that (tn) is convergent.Hint: Prove that ≤tn ≤ 1 for all n € N+ and use the previous exercise.(b) Compute the limit of (tn).

We have given a sequence tn=fnfn+1 where fj is the jth term of fibonacci sequence. We have to show…

= Consider the sequence defined by t, f, where f; is the jth term in the Fibonacci sequence (recall that f₁ = f2 = 1 and fa+1= fn + fa-1). (a) Prove that (t) is convergent. Hint: Prove that St $1 for all ne N+ and use the previous exercise.

= Consider the sequence defined by t, f, where f; is the jth term in the Fibonacci sequence (recall that f₁ = f2 = 1 and fa+1= fn + fa-1). (a) Prove that (t) is convergent. Hint: Prove that St $1 for all ne N+ and use the previous exercise.

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

Analysis

Consider the sequence defined by tn = ff, where f; is the jth term in the Fibonacci
sequence (recall that f₁ = f2 = 1 and fn+1 = fn + fn-1).
(a) Prove that (tn) is convergent.
Hint: Prove that ≤tn ≤ 1 for all n € N+ and use the previous exercise.
(b) Compute the limit of (tn).

Expert Solution

Step 1

We have given a sequence $t_{n} = \frac{f_{n}}{f_{n + 1}}$ where $f_{j}$ is the jth term of fibonacci sequence.

We have to show that sequence $t_{n}$ is convergent.

Result : A bounded and monotonic sequence is convergent.

We will prove that given sequence is bounded and monotonic .

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