
Concept explainers
(a)
To prove: The
(a)

Explanation of Solution
Using the mathematical induction and the Fibonacci recurrence.
The sequence of numbers
From our calculations, the first few terms of the Fibonacci sequence are
Using these values to calculate the next steps which is given in table 1.
n | |
0 | 0 |
1 | 1 |
2 | 3 |
3 | 8 |
4 | 21 |
5 | 55 |
6 | 144 |
7 | 377 |
n |
Table 1
Table 1 shows the recurrence relation for the given
Where, n is the numbers and
(b)
To prove: The
(b)

Explanation of Solution
Using the mathematical induction and the Fibonacci recurrence.
The sequence of numbers
From our calculations, the first few terms of the Fibonacci sequence are
Using these values to calculate the next steps which is given in table 2.
n | |
0 | 0 |
1 | 1 |
2 | 4 |
3 | 12 |
4 | 33 |
5 | 88 |
6 | 232 |
7 | 609 |
n |
Table 2
Table 2 shows the recurrence relation for the given
Where, n is the numbers and
(c)
To prove: The
(c)

Explanation of Solution
Using the mathematical induction and the Fibonacci recurrence.
The sequence of 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
Using these values to calculate the next steps which is given in table 3.
n | |
0 | |
1 | |
2 | 0 |
3 | |
4 | 1 |
5 | |
6 | 4 |
7 | |
n |
Table 3
Table 3 shows the recurrence relation for the given
Where, n is the numbers and
(d)
To prove:
(d)

Explanation of Solution
Using the mathematical induction and the Fibonacci recurrence.
The sequence of numbers
From our calculations, the first few terms of the Fibonacci sequence are
Using these values to calculate the next steps which is given in table 4.
n | |
0 | |
1 | |
2 | 2 |
3 | |
4 | |
5 | |
6 | |
7 | |
n |
Table 4
Table 4 shows the recurrence relation for the given
Where, n is the numbers and
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Chapter 7 Solutions
Introductory Combinatorics
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