OK, here is the solution in all the gory details.
Expanding the rightmost side, I get:
Now, look at P(k+1):
I am going to manipulate both sides of this statement, one at a time:
Isolate (k+1)st Term On Left Side:
Expand Right Side:
So, I have:
Now, remember P(k)? Assume it is true, and subtract it from P(k+1). It will be handy if you subtract the left side of P(k) from the left side of P(k+1), and do likewise with their right sides. I am going to denote both sides of P(k) with the color blue.
Then you get:
which is an identity. Therefore, P(k)-->P(k+1).
edit: fixed superscript bracket