Chapter 9.2, Problem 3E

Solution

a.

To generate the first ten terms of the Fibonacci sequence.

  $1,1,2,3,5,8,13,21,34,55,\mathrm{...}$

Given information:

The Fibonacci sequence.

Calculation:

The first two terms of the Fibonacci sequence are $F_1=1$ and $F_2=1$ .

The recursive definition for Fibonacci sequence is given as $F_n=F_{n-2}+F_{n-1}$ , where $F_1=1$ and $F_2=1$ .

Then, the first ten terms of the sequence can be written as follows:

  $\begin{array}{l}{F}_1=1\\ F_2=1\\ F_3=F_1+F_2=1+1=2\\ F_4=F_2+F_3=1+2=3\\ F_5=F_3+F_4=2+3=5\\ F_6=F_4+F_5=3+5=8\\ F_7=F_5+F_6=5+8=13\\ F_8=F_6+F_7=8+13=21\\ F_9=F_7+F_8=13+21=34\\ F_{10}=F_8+F_9=21+34=55\end{array}$

Therefore, the first ten terms of the Fibonacci sequence are as follows:

  $1,1,2,3,5,8,13,21,34,55,\mathrm{...}$

b.

To find the sum of the first ten terms of the Fibonacci sequence. Then divide the sum by 11 and notice the result.

The sum of the first ten terms is 143 and this sum is divisible by 11.

Given information:

The Fibonacci sequence.

Calculation:

The first ten terms of the Fibonacci sequence are $1,1,2,3,5,8,13,21,34,55,\mathrm{...}$ .

Add these terms to get the sum of first ten terms.

  $1+1+2+3+5+8+13+21+34+55=143$

Now, divide the sum by 11. This gives $\frac{143}{11}=13$ .

This shows that when the sum of the first ten terms of the Fibonacci sequence is divided by 11, it gives the seventh term of the sequence.

c.

To generate a Fibonacci like sequence using the first two terms other 1 and 1. Then divide the sum by 11 and notice the result.

A Fibonacci like sequence is $2,3,5,8,13,21,34,55,89,144,\mathrm{...}$ . The sum of the first ten terms is 374 and the sum is divisible by 11.

Given information:

Fibonacci like sequence using the first two terms other 1 and 1.

Calculation:

Consider the first two terms to be $F_1=2$ and $F_2=3$ .

Now, generate the Fibonacci like sequence using these two terms.

  $\begin{array}{l}{F}_1=2\\ F_2=3\\ F_3=F_1+F_2=2+3=5\\ F_4=F_2+F_3=3+5=8\\ F_5=F_3+F_4=5+8=13\\ F_6=F_4+F_5=8+13=21\\ F_7=F_5+F_6=13+21=34\\ F_8=F_6+F_7=21+34=55\\ F_9=F_7+F_8=34+55=89\\ F_{10}=F_8+F_9=55+89=144\end{array}$

Therefore, the first ten terms of the sequence are $2,3,5,8,13,21,34,55,89,144,\mathrm{...}$ .

Now, find the sum of the first ten terms of the sequence.

  $2+3+5+8+13+21+34+55+89+144=374$

Divide the sum by 11.

  $\frac{374}{11}=34$

This shows that when the sum of the first ten terms of the Fibonacci like sequence is divided by 11, it gives the seventh term of the sequence.

d.

To make a conjecture about the sum of the first ten terms of any Fibonacci like sequence.

The conjecture can be made is “The sum of the first ten terms of any Fibonacci like sequence is 11 times the seventh term of the sequence.”

Given information:

Any Fibonacci like sequence.

Calculation:

From part (b) and part (c), it is observed that the sum of the first ten terms of a Fibonacci sequence or Fibonacci like sequence when divided by 11, it equals the seventh term of the sequence.

That means, the sum of the first ten terms of the sequence is equal to 11 times the seventh term of the sequence.

Therefore, the conjecture can be made is “The sum of the first ten terms of any Fibonacci like sequence is 11 times the seventh term of the sequence.”