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aliina [53]
3 years ago
11

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

Computers and Technology
1 answer:
Vladimir [108]3 years ago
4 0

Explanation:

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

2^2+3^2=4+9=13

So 2^2+3^2 < 5^2 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

2^k+3^k

Our goal is then to show that 2^{k+1}+3^{k+1}.

We have that

2^{k+1}+3^{k+1}=2\cdot 2^k+3\cdot 3^k

(since we're assuming that 2^k+3^k, we know that 3\cdot (2^k+3^k) ).

So 2^{k+1}+3^{k+1}, and the inductive step is proven.

Therefore we can conclude by the principle of mathematical induction that for all n \geq 2,~2^n+3^n

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