Chapter 9.5, Problem 18E

Solution

To prove : using mathematical induction to prove that statement $3^n\ge 3n$ holds true for all positive integers.

Given information :

The statement $3^n\ge 3n$ .

Formula used :

Follow the steps to prove that the statement is true for any positive integer.

Anchor step: Prove that any statement $P_n$ is true for $n=1$ .

Inductive hypothesis: Assume $P_n$ is true for $n=k$ .

Inductive Step: Prove that $P_n$ is true for $n=k+1$ .

Proof :

Conjecture:

Take $P_n$ to be statement $3^n\ge 3n$ .

Anchor step:

Prove that $P_n$ is true for $n=1$ .

For $n=1$ ,

  $\begin{array}{l}{3}^1\stackrel{?}{=}3\cdot 1\\ 3=3\end{array}$

Inductive hypothesis:

Assume $P_n$ is true for $n=k$ . Replace $n$ with $k$ in $P_n$ .

  $P_k:3^k\ge 3k$

Inductive Step:

Prove that $P_n$ is true for $n=k+1$ . This means it is needed to be proved that $P_{k+1}$ must be true.

Proceed with the inductive hypothesis.

  $3^k\ge 3k$

Multiply both sides with $3$ .

  $\begin{array}{c}3\cdot 3^k\ge 3\cdot 3k\\ 3^{k+1}\ge 9k\end{array}$

Since $k\ge 1$ ,

  $\begin{array}{c}{3}^{k+1}\ge 9\left(k+1\right)\\ =3k+6k\\ \ge 3k+6\\ >3k+3\end{array}$

  $3^{k+1}\ge 3\left(k+1\right)$

The statement is same as the statement $P_{k+1}$ , so $P_{k+1}$ is true.

So, if $P_n$ is true for $n=k$ , then it is also true for $n=k+1$ .

Conclusion:

As $P_n$ is true for $n=1$ and $P_k$ further implies $P_{k+1}$ , this means $P_n$ is true for $n=2$ , $n=3$ and so on.

By mathematical induction, $P_n:3^n\ge 3n$ is true for all positive integers $n$ .