Thread: Fibonacci Numbers

Q: Show that the Fibonacci numbers F(1),F(2),F(3),... ,where F(1)= F(2)=1 and F(k)= F(k-2) + F(k-1) for k>2 satisfy the following equality for all n (greater or equal to) 1.

(F(1))^2 + (F(2))^2 + (F(3))^2+.....+ (F(n))^2 = F(n) * F(n+1)


I did the basis step:
1^2 + 1^2 = 1

but how do I do the inductive step for this question?



Thanks,
Creative

Originally Posted by Creative

Q: Show that the Fibonacci numbers F(1),F(2),F(3),... ,where F(1)= F(2)=1 and F(k)= F(k-2) + F(k-1) for k>2 satisfy the following equality for all n (greater or equal to) 1.
(F(1))^2 + (F(2))^2 + (F(3))^2+.....+ (F(n))^2 = F(n) * F(n+1)

I did the basis step:
1^2 + 1^2 = 1

You did not do the first step.

Is it true that

Oops basis step is 1^2(F1) = 1^2(F2) = 1?

1^2 +1^2 = F(2)*F(2+1)
is not true?
1^2*1^2? = 1

Originally Posted by Creative

Oops basis step is 1^2(F1) = 1^2(F2) = 1?
1^2 +1^2 = F(2)*F(2+1) is not true?

Yes it is true.

.

Thanks, so basis step would be proving F(1)
F(1)*(F(1+1)) = F(1)*F(2) = (1^2)*(1^2) = 1
so true for basis step.
Then how would I do inductive?

I wonder if you understand what the problem really says?
You want to prove .
We have more or less done the first step.
Now assume that is true.
On that basis prove it is true for .
I am not going to do it for you. For it is your problem.