I've been working on a proof by induction concerning the Fibonacci sequence and I'm stumped at how to do this.
Theorem: Given the Fibonacci sequence, $f_n$, then $f_{n+2}^2-f_{n+1}^2=f_nf_{n+3}$, $∀n∈N$
I have proved that this hypothesis is true for at least one value of $n$.
Consider $n=1$: $f_{1+2}^2-f_{1+1}^2=f_1f_{1+3}$
$f_{3}^2-f_{2}^2=f_1f_{4}$
$2^2-1^2=(1)(3)$
$3=3$
Consider $n=2$: $f_{2+2}^2-f_{2+1}^2=f_2f_{2+3}$
$f_{4}^2-f_{3}^2=f_2f_{5}$
$3^2-2^2=(1)(5)$
$9-4=(1)(5)$
$5=5$
I will assume that the hypothesis is true from $n=2$ up to some arbitrary value $k$: $f_{k+2}^2-f_{k+1}^2=f_kf_{k+3}$
and will prove true for $k+1$, showing that: $f_{k+3}^2-f_{k+2}^2=f_{k+1}f_{k+4}$
I know that $x^2-y^2=(x-y)(x+y)$, but the proof doesn't seem to work because $(f_{k+3}-f_{k+2})(f_{k+3}+f_{k+2})$ produces $(f_{k+1})(f_{k+5})$ instead of the expected $(f_{k+1})(f_{k+4})$.
How can I manipulate the proof to achieve the expected hypothesis?
Any suggestions you could provide would be greatly appreciated! Thank you!
Note that $f_{k+3} + f_{k+2} = f_{k+4}$.
Remember that when two consecutive Fibonacci numbers are added together, you get the next in the sequence. And when you take the difference between two consecutive Fibonacci numbers, you get the term immediately before the smaller of the two.
The sequence (in ascending order) goes $f_{k+1}, f_{k+2}, f_{k+3}, f_{k+4}$.
When you write it like that, it should be quite clear that $f_{k+3} - f_{k+2} = f_{k+1}$ and $f_{k+2} + f_{k+3} = f_{k+4}$.
Actually, you don't need induction. A direct proof using just that plus the factorisation (which you already figured out) is quite trivial (as long as you realise your error).