Chapter 13, Problem 1E

Compute the value of each of the following.

a. $F_{15}$

b. $F_{15}-2$

c. $F_{15-2}$

d. $F1515$

e. $F_{15/5}$

To determine

(a)

To calculate:

The value of $F_{15}$.

Answer to Problem 1E

Solution:

The value of $F_{15}$ is 610.

Explanation of Solution

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13 and so on, (each term is the sum of the first two preceding terms).

The terms of the Fibonacci sequence are known as Fibonacci numbers. The $Nth$ term of the Fibonacci sequence is denoted by $F_N$.

Given:

The given expression is $F_{15}$.

Formula used:

The recursive formula to calculate the $Nth$ Fibonacci number is given by,

$F_N=F_{N-1}+F_{N-2}$

Here $F_{N-1}$ and $F_{N-2}$ are the two preceding terms.

Calculation:

The $15\text{th}$ term in the sequence is the sum of $13\text{th}$ and $14\text{th}$ term. By using recursive formula,

$F_N=F_{N-1}+F_{N-2}$

Substitute 1 for $F_1$ and $F_2$ in the above recursive formula to calculate $F_3$,

$\begin{array}{rll}{F}_3& =& 1+1\\ & =& 2\end{array}$

Substitute 1 for $F_2$ and 2 for $F_3$ to calculate $F_4$,

$\begin{array}{rll}{F}_4& =& 1+2\\ & =& 3\end{array}$

Substitute 2 for $F_3$ and 3 for $F_4$ to calculate $F_5$,

$\begin{array}{rll}{F}_5& =& 2+3\\ & =& 5\end{array}$

Similarly, $F_6=8$, $F_7=13$, $F_8=21$, $F_9=34$, $F_{10}=55$, $F_{11}=89$, $F_{12}=144$, $F_{13}=233$, and $F_{14}=377$ are obtained.

Substitute $F_{13}$ and $F_{14}$ in the above recursive formula to calculate $F_{15}$.

$\begin{array}{rll}{F}_{15}& =& 233+377\\ & =& 610\end{array}$

Conclusion:

Thus, the $15\text{th}$ term in the sequence is 610.

To determine

(b)

To calculate:

The value of $F_{15}-2$.

Answer to Problem 1E

Solution:

The value of $F_{15}-2$ is 608.

Explanation of Solution

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13 and so on, (each term is the sum of the first two preceding terms).

The terms of the Fibonacci sequence are known as Fibonacci numbers. The $Nth$ term of the Fibonacci sequence is denoted by $F_N$.

Given:

The given expression is $F_{15}-2$.

Formula used:

The recursive formula to calculate the $Nth$ Fibonacci number is given by,

$F_N=F_{N-1}+F_{N-2}$

Here $F_{N-1}$ and $F_{N-2}$ are the two preceding terms.

Substitute 610 for $F_{15}$ from part (a) to calculate $F_{15}-2$,

$\begin{array}{rll}{F}_{15}-2& =& 610-2\\ & =& 608\end{array}$

Conclusion:

Thus, the value of $F_{15}-2$ is 608.

To determine

(c)

To calculate:

The value of $F_{15-2}$.

Answer to Problem 1E

Solution:

The value of $F_{15-2}$ is 233.

Explanation of Solution

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13 and so on, (each term is the sum of the first two preceding terms).

The terms of the Fibonacci sequence are known as Fibonacci numbers. The $Nth$ term of the Fibonacci sequence is denoted by $F_N$.

Given:

The given expression is $F_{15-2}$.

Formula used:

The recursive formula to calculate the $Nth$ Fibonacci number is given by,

$F_N=F_{N-1}+F_{N-2}$

Here $F_{N-1}$ and $F_{N-2}$ are the two preceding terms.

In the Fibonacci sequence $F_{15-2}$ is the $13\text{th}$ term of the sequence.

Substitute 89 for $F_{11}$ and 144 for $F_{12}$ from part (a) to calculate $F_{13}$,

$\begin{array}{rll}{F}_{13}& =& 89+144\\ & =& 233.\end{array}$

Conclusion:

Thus, the value of $F_{15}-2$ is 233.

To determine

(d)

To calculate:

The value of $\frac{F_{15}}{15}$.

Answer to Problem 1E

Solution:

The value of $\frac{F_{15}}{15}$ is 122.

Explanation of Solution

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13 and so on, (each term is the sum of the first two preceding terms).

The terms of the Fibonacci sequence are known as Fibonacci numbers. The $Nth$ term of the Fibonacci sequence is denoted by $F_N$.

Given:

The given expression is $\frac{F_{15}}{15}$.

Formula used:

The recursive formula to calculate the $Nth$ Fibonacci number is given by,

$F_N=F_{N-1}+F_{N-2}$

Here $F_{N-1}$ and $F_{N-2}$ are the two preceding terms.

Substitute 610 for $F_{15}$ from part (a) to get $\frac{F_{15}}{15}$,

$\begin{array}{rll}\frac{F_{15}}{15}& =& \frac{610}{15}\\ & =& 40.67\end{array}$

Conclusion:

Thus, the value of $\frac{F_{15}}{15}$ is $40.67$.

To determine

(e)

To calculate:

The value of $F_{15/5}$.

Answer to Problem 1E

Solution:

The value of $F_{15/5}$ is 2.

Explanation of Solution

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13 and so on, (each term is the sum of the first two preceding terms).

The terms of the Fibonacci sequence are known as Fibonacci numbers. The $Nth$ term of the Fibonacci sequence is denoted by $F_N$.

Given:

The given expression is $F_{15/5}$.

Formula used:

The recursive formula to calculate the $Nth$ Fibonacci number is given by,

$F_N=F_{N-1}+F_{N-2}$

Here $F_{N-1}$ and $F_{N-2}$ are the two preceding terms.

The number $F_{15/5}$ is the third term of the sequence given by,

$\begin{array}{rll}{F}_{15/5}& =& F_3\\ & =& F_2+F_1\\ & =& 1+1\\ & =& 2\end{array}$

Conclusion:

Thus, the value of $F_{15/5}$ is 2.