Assuming F represents the Fibonacci sequence, use mathematical induction to prove that for all integers n >= 0, Fsub(n+2)Fsub(n) − (Fsub(n+1))^2 = (−1)^n.
I have gotten to the inductive goal when plugging in k+1, but am lost from here.
Thanks.
Assuming F represents the Fibonacci sequence, use mathematical induction to prove that for all integers n >= 0, Fsub(n+2)Fsub(n) − (Fsub(n+1))^2 = (−1)^n.
I have gotten to the inductive goal when plugging in k+1, but am lost from here.
Thanks.
Assuming, (after confirming base case)
F_{k+2}F_{k} − (F_{(k+1)}^{2} = (−1)^{k}. (Assumption P)
With the induction step, you want to prove
F_{k+3}F_{k+1} - (F_{k+2})^{2} = (-1)^{k+1}
so,
(-1)^{k+1} = (-1)^{k}*(-1)^1 = -1 * ( F_{k+2}F_{k} − (F_{(k+1)}^{2} ) (by Assumption P)
Can you take it from here? I currently don't have pencil and paper handy to give you the full answer.
You'll probably also have to use the fact that F_{k+2} = F_{k} + F_{k+1}
For simplicity, I would leave the bolded and then work in the other direction as well
In other words, working with F_{k+3}F_{k+1} - (F_{k+2})^{2} so that it's equal to what is in the bold, since A = B = C implies A = C