Induction to prove $2n + 3 < 2^n$

Question

I am having trouble and was wondering if someone could go over the steps slowly to show that:

$$2n + 3 < 2^n \ \text{for} \ n \geq 4$$

Any help would be amazing!

Answer 1

Base case: Just check it when $n=4$.

Inductive step: Suppose $2n+3 < 2^n$ for some $n$. Then it must be the case that $$2(2n+3) < 2 \times 2^n$$But what does this tell you?

Hint: Compare the left-hand side of this inequality with $2(n+1)+3$.

Answer 2

If $2^m>2m+3$

$2^{m+1}-2(m+1)-3>2(2m+3)-2(m+1)-3=4m+1>0$

Now, for $m=1,2^m-(2m+3)=2^1-(2+3)<0$

$$\cdots$$

For $m=3,2^m-(2m+3)=2^4-(2\cdot3+3)=0$

For $m=4,2^m-(2m+3)=2^4-(2\cdot4+3)>0$

Answer 3

For the induction step, if $2n+3<2^n$ (and $n\ge 0$) then $$2(n+1)+3 = 2n+5 <4n+6=2(2n+3)<2\cdot 2^n=2^{n+1}.$$