Induction Help: prove $2n+1< 2^n$ for all $n$ greater than or equal to $3$.
I understand that he expanded the left side but I'm having trouble figuring out what he did on the right side of the inequality. Where the did the $2$ (in circle) come from?
Solution 1:
In the induction hypothesis, it was assumed that $2k+1 < 2^k,\forall k \geq 3$, So when you have $2k + 1 +2$ you can just sub in the $2^k$ for $2k+1$ and make it an inequality. So that makes $$2k+1+2 < 2^k + 2$$ and since it was assumed $k\geq 3$ we also know that $2 < 2^k$. So now we have $$2k+1+2 < 2^k+2 < 2^k+2^k=2^{k+1}$$. Then by the transitive property we have $$2(k+1)+1 < 2^{k+1}$$