comes directly from the definition of the Fibonacci numbers. Remember that ?
So in your case in
and
,
since
I am trying to figure out how to show that when n is a positive integer.
My Work
P(n) is for
Basis step - P(1) is true because
Inductive step - Assume is true,
then is true
Then prove , but I can't figure out how to get this P(k+1) equation to equal
I received the reply below, which is probably a wonderful answer, but I am unclear as to where in the 3rd line came from. I though you had to add the initial part of the P(k+1) equation, which is , to both sides of the P(k) equation as I did above.
First Reply
Assume for n.
We want to show that
,<--- negative of the induction hypothesis
by the induction hypothesis.
So, is true.
The math makes sense but I still do not understand why you are allowed to do the "negative of the inductive hypothesis" to get the extra -1 to make the answer . Doesn't the answer also still need to be ?
Also, I thought it wasn't supposed to prove the exact P(k+1) equation in the inductive step but rather the P(k) equation with the first part of P(k+1) added to both sides as follows:
Then, aren't I supposed to work on the second half of this equation to make it equal ?
EDIT:
The math makes sense but I still do not understand why you are allowed to do the "negative of the inductive hypothesis" to get the extra -1 to make the answer be ? I already understand that - I just don't know where the exra -1 came from.