\begin{align*} (k+1)^2 \amp = k^2 + 2k + 1 \amp\\ \amp \lt 2^k + 2k + 1 \amp \ldots\text{by the inductive hypothesis.}\\ \amp \lt 2^k + 2^k \amp \ldots\text{ since } 2k + 1 \lt 2^k \text{ for }k \ge 5.\\ \amp = 2^{k+1}. \amp \end{align*}
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