Chapter 3.5, Problem 31PPS

a.

To determine

The relationship between the terms of the sequence and find the next five terms in the sequence.

Answer to Problem 31PPS

  $13,21,34,55,89$

Explanation of Solution

Given:

The Fibonacci sequence can be defined by a recursive formula. The first six terms are 1,1,2,3,5,8,….

Calculation:

The Fibonacci sequence is given by 1,1,2,3,5,8,…

Here observe that starting from the third term onward, each term is the sum of the previous two terms.

So, in Fibonacci sequence, each term is the sum of the previous two terms in the sequence.

So, in order to find the next five terms, add the preceding two terms for each. Like the next term would be

  $5+8=13$

Similarly, the next term would be $8+13=21$ .

In the same way the next three terms would be 34, 55 and 89.

b.

To determine

The formula for the nth term if $n\ge 3$ .

Answer to Problem 31PPS

  $a_n=a_{n-2}+a_{n-1}$

Explanation of Solution

Given:

The Fibonacci sequence can be defined by a recursive formula. The first six terms are 1,1,2,3,5,8,….

Calculation:

Any term in the Fibonacci sequence is given by the sum of the previous two terms.

Let $a_n$ be the nth term of the sequence with $n\ge 3$ . So, the nth term formula is given by

  $a_n=a_{n-2}+a_{n-1}$

c.

To determine

The 15^th term.

Answer to Problem 31PPS

  $610$

Explanation of Solution

Given:

The Fibonacci sequence can be defined by a recursive formula. The first six terms are 1,1,2,3,5,8,….

Calculation:

The 15^th term of the sequence will be the sum of the 13^th and 14^th terms of the sequence.

First 14 terms of the sequence are $1,1,2,3,5,8,13,21,34,55,89,144,233,377$ .

So, the 15^th term will be the sum of 233 and 377. That is, the 15^th term of the sequence is

  $233+377=610$

Thus, the 15^th term of the sequence is 610.

d.

To determine

To explain why the Fibonacci sequence is not an arithmetic sequence.

Explanation of Solution

Given:

The Fibonacci sequence can be defined by a recursive formula. The first six terms are 1,1,2,3,5,8,….

Calculation:

The common difference between any two terms of the arithmetic sequence is always equal, while the difference between any two terms of the Fibonacci sequence is not same for any two terms.

Thus, the Fibonacci sequence is not an arithmetic sequence.