I need to prove $2n \leq 2^n$, for all integer $n≥1$ by mathematical induction?
This is how I prove this:
Prove:$2n ≤ 2^n$, for all integer $n≥1$
Proof: $2+4+6+...+2n=2^n$
$i.)$ Let $P(n)=1 P(1): 2(1)=2^1\implies 2=2$. Hence, $P(1)$ is true.
$ii.)$ Assume that $P(n)$ is true for $n=k$, i.e, $2+4+6+...+2k=2^k$, and prove that $P(n)$ is also true for $n=k+1$, i.e, $2+4+6...+2(k+1)=2^{(k+1)}$
from the assumption add $2(k+1)$ on both sides so we have $2+4+6...2k+2(k+1)=2^k+2(k+1)$
I'm confused with $2^k+2(k+1)$, I don't know how to make $2^k$ be equivalent to $2^{k+1}$. I feel i'm doing something wrong.
Any help would be appreciated!