Prove by induction for $\forall n\in \Bbb N_1 $ that $2^n \ge 1+n $.

Question

Can someone please help me in this exercise?

$$2^n \ge 1+n $$

What I've done so far is prove for $n=1$. So:

$$2^1 \ge 1+1 $$ $$2 \ge 2 $$ Which is true

Now, I'm supposing that what I have to prove is for $n+1$:

$$2^{n+1} \ge 1+ (n+1)$$

But I don't know how to continue it.

Thank you for the help

Answer 1

Simply note that

$$2^{n+1} =2\cdot 2^n\stackrel{ind. hyp.}\ge 2(n+1)=1+(n+1)+n\ge 1+ (n+1)$$