The sequence of numbers $1,1,2,3,5,8,13,21,34,55, \ldots$. is called the Fibonacci sequence. It can be generated by setting $f_0=1, f_1=1$ and setting $f_n=f_{n-1}+f_{n-2}$ for $n \geq 2$. Notice that all numbers in the sequence are integers.
The following formulas all relate to the definition of the Fibonacci sequence. You will need to remember to use your inductive hypothesis in each case. You will also need to use the definition of the sequence: $f_n=f_{n-1}+f_{n-2}$ for $n \geq 2$.
(a) Prove that $f_0^2+f_1^2+f_2^2+\cdots+f_n^2=f_n f_{n+1}$ for $n \geq 0$.
(b) Show that $f_0+f_2+\cdots+f_{2 n}=f_{2 n+1}$ for $n \geq 0$.
(c) Prove that $f_{n-1} f_{n+1}-f_n^2=(-1)^{n+1}$ for $n \geq 1$.
(d) It is an amazing fact that $f_n$, the $n$th element of the sequence, can be given by a formula that involves $\sqrt{5}$. The formula is given by
$$
f_n=c_1 a^n+c_2 b^n,
$$
where $n \geq 0$ and
$$
c_1=\frac{1+\sqrt{5}}{2 \sqrt{5}} \quad c_2=\frac{-(1-\sqrt{5})}{2 \sqrt{5}}
$$
and
$$
a=\frac{1+\sqrt{5}}{2} \quad b=\frac{1-\sqrt{5}}{2} .
$$
It is worth mentioning that $a$ and $b$ are the two solutions to the equation $x^2=x+1$.
Use complete induction to prove that the formula is correct.