Prove the following statements with either induction, strong induction or proof by smallest counterexample. Concerning the Fibonacci sequence, prove that $\sum_{k=1}^{n} F_{k}^{2}=F_{n} F_{n+1}$.

Show that the Strong Form of Mathematical Induction and the form of mathematical induction where the Inductive Step is "if S(n) is true, then S(n+1) is true" are equivalent. That is, assume the Strong Form of Mathematical Induction and prove the alternative form; then assume the alternative form and prove the Strong Form of Mathematical Induction.

Consider the Fibonacci sequence $\left\{a_n\right\}$. (a) Use mathematical induction to prove that

$$a_{n+1} a_{n-1}=\left(a_n\right)^2+(-1)^n .$$

(b) Use mathematical induction to prove that

$$a_n=\frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}} .$$

$F_n$ denotes the $n$th term of the Fibonacci sequence discussed in the Section earlier. Use mathematical induction to prove the statement.

$$F_1+F_2+F_3+\cdots+F_n=F_{n+2}-1$$

(a) What is a recursively defined sequence?

(b) What is the Fibonacci sequence?

What is wrong with the following argument, which purports to prove that all the Fibonacci numbers after the first two are even?

Let $f_n$ denote the nth term of the Fibonacci sequence. We prove that $f_n$ is even for all $n \geq 3$ using the strong form of the Principle of Mathematical Induction. The Fibonacci sequence begins 1, 1, 2. Certainly, $f_3=2$ is even and so the assertion is true for $n_0=3$. Now let k > 3 be an integer and assume that the assertion is true for all $n, 3 \leq n<k$; that is, assume that $f_n$ is even for all n < k. We wish to show that the assertion is true for n = k, that $f_k$ is even. But $f_k=f_{k-1}+f_{k-2}$. Applying the induction hypothesis to k - 1 and to k - 2, we conclude that each of $f_{k-1}$ and $f_{k-2}$ is even, hence, so is the sum. By the Principle of Mathematical Induction, $f_n$ is even for all $n \geq 3$.

Let f1, f2, .... fn, ... be the Fibonacci sequence. Use mathematical induction to prove that f1 + f2 + . . . +fn = f n+2 - 1

F_n denotes the nth term of the Fibonacci sequence discussed in Section 9.1. Use mathematical induction to prove the statement.

$F_{3 n}$ is even for all natural numbers $n$.

$F_n$ denotes the $n$th term of the Fibonacci sequence discussed in the Section earlier. Use mathematical induction to prove the statement.

$F_{3 n}$ is even for all natural numbers $n$.

Let $y _ { 1 } = 6 ,$ and for each $n \in \mathrm { N }$ define $y _ { n + 1 } = \left( 2 y _ { n } - 6 \right) / 3$ (a) Use induction to prove that the sequence satisfies $y _ { n } > - 6$ for all $n \in N$ (b) Use another induction argument to show the sequence $\left( y _ { 1 } , y _ { 2 } , y _ { 3 } , \ldots \right)$ is decreasing.

Any number in the Fibonacci sequence is a Fibonacci number.

Name three Fibonacci numbers that are composite numbers.

Any number in the Fibonacci sequence is a Fibonacci number.

In the first ten terms of the Fibonacci sequence, which ones are odd numbers?

Use the principle of mathematical induction to prove that for the Fibonacci numbers $F_k$,

$$F_1+F_2+F_3+\cdots+F_n=F_{n+2}-1$$

Examples of the Fibonacci sequence in nature.