1.
1^3 + 2^3 + ... + n3 = (n(n+1)/2 )^2 for any positive integer n.
2.
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,
Note that:
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.