Induction and Fibonacci Numbers

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 http://www.mathhelpforum.com/math-he...dda283da-1.gif 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 http://www.mathhelpforum.com/math-he...63f299fa-1.gif for n.

We want to show that http://www.mathhelpforum.com/math-he...c1962bf5-1.gif

http://www.mathhelpforum.com/math-he...dda283da-1.gif

http://www.mathhelpforum.com/math-he...e5104150-1.gif

http://www.mathhelpforum.com/math-he...b4994ac0-1.gif,<--- negative of the induction hypothesis

http://www.mathhelpforum.com/math-he...78c4ac77-1.gif by the induction hypothesis.

http://www.mathhelpforum.com/math-he...f39e74ff-1.gif

So, http://www.mathhelpforum.com/math-he...63f299fa-1.gif is true.