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Q 92.

Precalculus Enhanced with Graphing Utilities
Found in: Page 810
Precalculus Enhanced with Graphing Utilities

Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

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Short Answer

Fibonacci Sequence: Let


define the nth term of a sequence.

(a) Show that u1= 1 and u2 = 1.

(b) Show that un+2 = un+1 + un.

(c) Draw the conclusion that {un} is the Fibonacci sequence.

(a) The first (u1) and second (u2) term of the sequence are:

u1=1 & u2=1

(b) un+2=un+1+un

(c) {un } is a Fibonacci series.

See the step by step solution

Step by Step Solution

Step 1. Write the given information.

The nth term of the sequence is:


Step 2. Use the nth sequence formula to calculate first and second term.

First term is:


Second term is:


Step 3. To prove this expression, compute both sides of the expression to be equivalent. 

Firstly, right hand side:

un+2=(1+5)n+2-(1-5)n+22n+25un+2=(1+5)n(1+5)2-(1-5)n(1-5)222 ×2n5un+2=(1+5)n1+5+25-(1-5)n1-25+54 ×2n5un+2=2×{(1+5)n3+5-(1-5)n3-5}4 ×2n5un+2={(1+5)n3+5-(1-5)n3-5}2 ×2n5

Now, the left hand side:


Since both the sides have equal results, the expression stands valid.

Step 4. Use the expression from Step 3.

As proved in Step 3, the expression is a form of Fibonacci sequence. Thus, {un} is a Fibonacci sequence.

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