1^3 + 2^3 + ... + n3 = (n(n+1)/2 )^2 for any positive integer n.
3^n < n! for all n > 6 with n ∈ N.
mathematical induction to prove these?
Yes that's the way to go. Without going through all the formalities ...
#1: Assume it holds for . We want to show that it holds for as well, that is, we want to show that
So for your inductive step:
#2: Again, assume it holds for for all , so we're assuming that . We then want to show that it is true for as well, that is,
We can assume the blue because we're given that the statement is true for values strictly greater than 6. So obviously, . Hopefully you'll see the conclusion directly after using the inductive step.