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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

SKILLS Plus

Formula Using Fibonacci Numbers Let a n be the nth term of the sequence defined recursively by

a n + 1 = 1 1 + a n

and let a 1 = 1 . Find a formula for a n in the terms of the Fibonacci numbers F n . Prove that the formula you found is valid for all natural numbers n .

To determine

To find:

A formula for an in the terms of the Fibonacci numbers Fn, is valid for all natural numbers n.

Explanation

Given:

an be the nth term of the sequence. It is defined recursively as

an+1=11+an

and a1=1

Approach:

The Fibonacci numbers F0, F1, F2, are defined as follows,

F0=0

F1=1

Fn=Fn1+Fn2 (n2)

Here, n denotes natural numbers.

Calculation:

The Fibonacci sequence is given by,

F2=F1+F0=1

F3=F2+F1=2

F4=F3+F2=3

F5=F4+F3=5

F6=F5+F4=8

F7=F6+F5=13

F8=F7+F6=21

The nth term of a sequence is defined as follows,

an+1=11+an

So, the terms of this sequence are as follows,

a1=1

a2=11+a1=12=F2F3

a3=11+a2=23=F3F4

a4=11+a3=35=F4F5

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