Question

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.

$n^3+2n$ is divisible by$3$

Short answer

The given statement is shown.

Step-by-step solution

Step 1. Given information

The given expression is$n^3+2n$

Step 2. Show that the given statement is correct.

First we consider the statement for $n=1$

Because $1^3+2.1=3$is divisible by 3 , the statement is true for width="37">n=1

Now we consider that the statement holds for some $k$

We need to show that $(k+1{)}^3+2(k+1)$is divisible by 3

width="305">(k+1)3+2(k+1)=k3+3k2+3k+1+2k+2=k3+2k+3k2+3k+3=k3+2k+3k2+k+1

Since $3\left(k^2+k+1\right)$is divisible by 3 and $k^3+2k$is divisible by 3 , it follows that $(k+1{)}^3+2(k+1)$is divisible by 3.

As a result statement 1 is true for all natural numbers$n$