Hi,

Could someone help me with setting up a problem . So i want to prove something by induction on k = 0,1,2,3,4...n, clearly $\displaystyle k \in Z+ $

I have two functions: g(x) and f(x) and i have observed that

$\displaystyle g(0) = y $

$\displaystyle f(0) = g(0) = y $

$\displaystyle f(1) = g(f(0))+g(g(f(0))) $

$\displaystyle f(2) = g(f(1))+g(g(f(1)))$

$\displaystyle ...$

so it looks like :

$\displaystyle f(k) = g(f(k-1))+g(g(f(k-1)))$ and if i check that for all k up to 100 i get correct numbers.

And as I said i would like to prove this ($\displaystyle f(k) = g(f(k-1))+g(g(f(k-1)))$) by induction on k So my question is what is my base case and what my induction hypothesis? To me it looks like my base case is :

$\displaystyle g(0) = y $

$\displaystyle f(0) = g(0) = y $

and my induction H:

$\displaystyle f(k) = g(f(k-1))+g(g(f(k-1)))$

but what confuses me is if if i plug in 0 for k here i get :

$\displaystyle f(0) = g(f(0))+g(g(f(0)))$

$\displaystyle f(0) = g(y)+g(g(y))$

which is not y which means that either i have wrong idea about my base case (although i always recieve f(0) = g(0)) or my induction hypothesis is wrong (but for all k > 0 i always get the right numbers if i apply $\displaystyle f(k) = g(f(k-1))+g(g(f(k-1)))$ ). I don't know how to connect these two together.

One more information . This ofcourse works if y = 0 but y = 1,2,3..n so $\displaystyle y\in N$ and $\displaystyle g(y)> y$

Thank you

baxy