How much of this have you done?
Have you done any of it?
If you have, what pattern do you see?
I'm doing part c now,
Basis is easy when k = 3.
Inductive step
Let x >=3 be an integer.
Our IH is (from the formula's I found for part b)
f^x (n) = n + x when x is even, and
f^x (n) = n + x + 2(-1)^n when x is odd
We want to prove that
f^x+1(n)=n+x+1 when x+1 is even, and
f^x+1(n)=n+x+1+2(-1)^n when x+1 is odd
Is the above correct?
So we have two cases from here:
Case 1: when x+1 is even, x is odd
Now, f^x+1(n) = f(f^x(n)) = f(n +x + 2(-1)^n)
Now, I think we have two more sub cases but I'm not sure?
Case 2: when x+1 is odd, x is even
Now, f^x+1(n) = f(f^x(n)) = f(n + x)
Now, I think we have two more sub cases but I'm not sure?
Can someone help me finish the sub cases?
Thanks!