<s>[INST] Use induction to prove $2n + 1 \le 2^n$ for $n=3,4,\ldots$ Use induction to prove $2n + 1 \le 2^n$ for $n=3,4,\ldots$
I've plugged $3$ in for $n$
I get $7 \le 8$
then I set $2(n+1) +1 \le 2^{n+1}$
then I'm lost.
 [/INST] We want to show that if $2k+1\le 2^k$ for a particular integer $k\ge 3$,  then $2(k+1)+1\le 2^{k+1}$.
We have $2(k+1)+1=2k+1 +2$. By the induction hypothesis, $2k+1\le 2^k$, and therefore $2k+1+2 \le 2^k+2$. 
But if $k\ge 1$, then $2^k+2\le 2^k+2^k=2^{k+1}$, so we are finished. 
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