Proof: Now (n+1)^2 = n^2+ 2n+1 and because n^2 < 2^n, it follows that n^2 + 2n + 1 < 2^n + 2n + 1. Finally, 2n+1 < 2^n and so (n+1)^2 = n^2 + 2n + 1 < 2^n + 2^n = 2^(n+1)
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