Question

using mathematical induction ,prove 5^n +6^n<9^n for n>=2

Answer 1

is it true for n=2?

5^2 + 6^2 = 61
9^2 = 81
so, yes.

Now, assuming it's true for n=k, what about n=k+1?
5^(k+1) + 6^(k+1)
= 5*5^k + 6*6^k
= 6*5^k + 6*6^k - 5^k
= 6(5^k+6^k) - 5^k
< 6*9^k - 5^k < 9*9^k = 9^(k+1)
so, yes.