Question

For which nonnegative integer’s n is $2n+3\le 2^n?$Prove your answer.

Short answer

P(n) is true for all integers n>3.

Step-by-step solution

Step: 1

The intersection of these two graphs is (3.247, 9.494).

Since n is an integer :n>3.

If n=4,

$\begin{array}{r}2\left(4\right)+3\le 2^4\\ 11\le 16\end{array}$

it is true for n=4.

Step: 2

Let P(k) be true.

$2k+3\le 2^k$

We need to prove that P(k+1) is true.

Step: 3

$\begin{array}{r}2(k+1)+3=(2k+3)+2\\ \le 2^k+2\\ =2\left(2^{k-1}+1\right)\\ \le 2\left(2^k\right)\\ =2^{k+1}\end{array}$

It is true for P(k+1) is true.