I have established the relation that
but how do you deliver a proof by induction for 2 variables???
I can't say I understand the relation, but if you're trying to prove something of the form: For all n and k, R(n,k) where R(n,k) would be the sentence that you have in your picture (i.e. the expression above "n = 1,2,3,..."); then you can prove that:
1) The sentence R(1,1) is true
2) If the sentence R(1,k') is true then R(1,k'+1) is true
3) Conclude that for all k, R(1,k) is true
4) Show that if, for all k, R(n',k) is true then for all k, R(n'+1,k) is true
5) Conclude that for all n, and for all k, R(n,k) is true.
Separate names with a comma.