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## Homework Statement

"Prove: ##∀n∈ℕ##, ##3^n>n^2##

## Homework Equations

## The Attempt at a Solution

(1) We will prove that ##3^n>n^2## at ##n=1##

##3=3^1>1=1^2##

(2) Now assume that ##3^k>k^2## for some ##k>1##

(3) We will prove that ##3^{k+1}>(k+1)^2## or ##3⋅3^k>k^2+2k+1##

Note that ##k^2+2k^2+1=3k^2+1≥k^2+2k+1##, and as such, ##3⋅3^k≥3k^2+3>3k^2+1≥k^2+2k+1##. So ##3^{k+1}>3k^2+1≥(k+1)^2##.

I'm basically ambivalent as to whether to insert that extra term in the inequality: ##3k^2+1##, not knowing whether it is equal to, less than, or greater than ##3^{k+1}##. Can someone check my work, please? Thanks.

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