Use induction on n to prove that fir all n> =2, 2^n+3^n< 5^n

Explanation:

We first prove the base case, which is proving that the inequality holds for n=2:

So and base case is proven.

We then do the inductive step, which is assuming that the inequality holds for n=k, and proving out of that that the inequality also holds for n=k+1:

Assume the inequality holds for n=k. This means that

Our goal is then to show that .

We have that

(since we're assuming that , we know that ).

So , and the inductive step is proven.

Therefore we can conclude by the principle of mathematical induction that for all

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