Chapter 7, Problem 1E

(a)

To determine

To prove: The $f_1+f_3+\mathrm{...}+f_{2n-1}$ using the mathematical induction and the Fibonacci recurrence relation.

Explanation of Solution

Using the mathematical induction and the Fibonacci recurrence.

The sequence of numbers $f_0,f_1,f_2,\mathrm{...},f_n,\mathrm{...}$ satisfying the recurrence relation and initial conditions are $f_n=f_{n-1}+f_{n-2}$ where, $n\ge 2$ is called the Fibonacci sequence, and the terms of the sequence are called Fibonacci numbers. The recurrence relation in (7.4) is also called the Fibonacci recurrence.

From our calculations, the first few terms of the Fibonacci sequence are

$0,1,1,2,3,5,8,13,21,34,\mathrm{...}$.

Using these values to calculate the next steps which is given in table 1.

n	${\displaystyle {\sum }_{k=1}^n{f}_{2k-1}}$
0	0
1	1
2	3
3	8
4	21
5	55
6	144
7	377
n	$f_{2n}$

Table 1

Table 1 shows the recurrence relation for the given $f_1+f_3+\mathrm{...}+f_{2n-1}$.

Where, n is the numbers and ${\displaystyle {\sum }_{k=1}^n{f}_{2k-1}}$ is the mathematical induction series.

(b)

To determine

To prove: The $f_0+f_2+\mathrm{...}+f_{2n}$ using the mathematical induction and the Fibonacci recurrence.

Explanation of Solution

Using the mathematical induction and the Fibonacci recurrence.

The sequence of numbers $f_0,f_1,f_2,\mathrm{...},f_n,\mathrm{...}$ satisfying the recurrence relation and initial conditions are $f_n=f_{n-1}+f_{n-2}$ where, $n\ge 2$ is called the Fibonacci sequence, and the terms of the sequence are called Fibonacci numbers. The recurrence relation in (7.4) is also called the Fibonacci recurrence.

From our calculations, the first few terms of the Fibonacci sequence are

$0,1,1,2,3,5,8,13,21,34,\mathrm{...}$.

Using these values to calculate the next steps which is given in table 2.

n	${\displaystyle {\sum }_{k=0}^n{f}_{2k}}$
0	0
1	1
2	4
3	12
4	33
5	88
6	232
7	609
n	$f_{2n+1}-1$

Table 2

Table 2 shows the recurrence relation for the given $f_0+f_2+\mathrm{...}+f_{2n}$.

Where, n is the numbers and ${\displaystyle {\sum }_{k=0}^n{f}_{2k}}$ is the mathematical induction series.

(c)

To determine

To prove: The $f_0-f_1+f_2-\mathrm{...}+{\left(-1\right)}^n{f}_n$ using the mathematical induction and the Fibonacci recurrence.

Explanation of Solution

Using the mathematical induction and the Fibonacci recurrence.

The sequence of numbers $f_0,f_1,f_2,\mathrm{...},f_n,\mathrm{...}$ satisfying the recurrence relation and initial conditions are $f_n=f_{n-1}+f_{n-2}$ where, $n\ge 2$ is called the Fibonacci sequence, and the terms of the sequence are called Fibonacci numbers.

The recurrence relation in (7.4) is also called the Fibonacci recurrence.

From our calculations, the first few terms of the Fibonacci sequence are

$0,1,1,2,3,5,8,13,21,34,\mathrm{...}$.

Using these values to calculate the next steps which is given in table 3.

n	${\displaystyle {\sum }_{k=0}^n{\left(-1\right)}^n{f}_k}$
0	$0$
1	$-1$
2	0
3	$-2$
4	1
5	$-4$
6	4
7	$-9$
n	$-1+{\left(-1\right)}^n{f}_{n-1}$

Table 3

Table 3 shows the recurrence relation for the given $f_0-f_1+f_2-\mathrm{...}+{\left(-1\right)}^n{f}_n$.

Where, n is the numbers and ${\displaystyle {\sum }_{k=0}^n{\left(-1\right)}^n{f}_k}$ is the mathematical induction series.

(d)

To determine

To prove: $f_0^2+f_1^2+\mathrm{...}+f_n^2$ by using the mathematical induction and the Fibonacci recurrence.

Explanation of Solution

Using the mathematical induction and the Fibonacci recurrence.

The sequence of numbers $f_0,f_1,f_2,\mathrm{...},f_n,\mathrm{...}$ satisfying the recurrence relation and initial conditions are $f_n=f_{n-1}+f_{n-2}$ where, $n\ge 2$ is called the Fibonacci sequence, and the terms of the sequence are called Fibonacci numbers. The recurrence relation in (7.4) is also called the Fibonacci recurrence.

From our calculations, the first few terms of the Fibonacci sequence are

$0,1,1,2,3,5,8,13,21,34,\mathrm{...}$.

Using these values to calculate the next steps which is given in table 4.

n	${\displaystyle {\sum }_{k=0}^n{f}_k{}^2}$
0	$0$
1
2	2
3	$6=2\times 3$
4	$15=3\times 5$
5	$40=5\times 8$
6	$104=13\times 21$
7	$273=13\times 21$
n	$f_n{f}_{n+1}$

Table 4

Table 4 shows the recurrence relation for the given $f_0^2+f_1^2+\mathrm{...}+f_n^2$.

Where, n is the numbers and ${\displaystyle {\sum }_{k=0}^n{f}_k^2}$ is the mathematical induction series.