Chapter 11.4, Problem 7E

To determine

To Prove: $\left(n^3+2n\right)$ is multiple of 3.

Answer to Problem 7E

It is prove that, by the Principle of Mathematical Induction $P\left(n\right)$ is true for all natural numbers n .

Explanation of Solution

Given information: $\left(n^3+2n\right)$

Let the statement $P\left(n\right)$ given as

  $P\left(n\right)$ : $\left(n^3+2n\right)$ is a multiple of 3, for every natural number n .

We observe that $P\left(1\right)$ is true,

Since $\left(1^3+2\times 1\right)=\left(1+2\right)=3$ ×1, is a multiple of 3.

Assume that $P\left(n\right)$ is true for some natural number k ,

Therefore, $P\left(k\right)$ : $\left(k^3+2k\right)$ , is a multiple of 3,

Hence, $\left(k^3+2k\right)=3q$ , where $q\in N$

Now, to prove that $P\left(k+1\right)$ is true, we have

  $\begin{array}{l}P\left(k+1\right):\left\{{\left(k+1\right)}^3+2\left(k+1\right)\right\}\\ =\left(k+1\right)\left(k^2+2k+1+2\right)\\ =\left(k+1\right)\left(k^2+2k+3\right)=3m,\text{where }m\in N\end{array}$

Thus $P\left(k+1\right)$ is true, whenever $P\left(k\right)$ is true.

Hence, by the Principle of Mathematical Induction $P\left(n\right)$ is true for all natural numbers n .