Assume for n.
We want to show that
,<--- negative of the induction hypothesis
by the induction hypothesis.
So, is true.
I am trying to figure out how to show that f_(n+1)f_(n-1)-(f_n)^2=(-1)^n when n is a positive integer.
My Work
P(n) is f_(n+1)f_(n-1)-(f_n)^2=(-1)^n for n>=1
Basis step - P(1) is true because f_2f_0-(f_1)^2=1*0-1=-1=(-1)^1
Inductive step - Assume P(k)=f_(k+1)f_(k-1)-(f_k)^2=(-1)^k is true, then P(k+1)=f_(k+2)f_(k)-(f_(k+1))^2=(-1)^(k+1) is true
P(k+1)=[f_(k+1)f_(k-1)-(f_k)^2]+[f_(k+2)f_(k)-(f_(k+1))^2]=(-1)^k+[f_(k+2)f_(k)-(f_(k+1))^2]
I can't figure out how to get this equation to equal (-1)^(k+1)
I am having such a hard time with this problem but you definitely opened up my eyes to a totally new approach. Thank you sooo much. I understand all the math but I am not totally sure where you came up with the following:
I'm sure its obvious, I have just gotten myself so confused. Do you add the k+1 equation
to both sides of ? I was trying to teach myself by looking at examples but it doesn't seem to be working out.
Thank you! I'm concentrating so hard that I'm missing the obvious!
You simply switched the terms around to get an extra negative on the right side of the equation? and you simplified the P(k+1) equation which made it the negative of the P(k) equation, correct?
Also, am I incorrect to think you have to make the following equation equal , not just the ?: