Question

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

$n(n+1)(n+2)$is divisible by$6$

Short answer

The given statement is shown.

Step-by-step solution

Step 1. Given information

The expression is$n(n+1)(n+2)$

Step 2. Show that the given statement is correct.

First we consider the statement for $n=1$

Because $1(1+1)(1+2)=6$is divisible by 6 , the statement is true for width="37">n=1

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

We need to show that $(k+1)(k+2)(k+3)$is divisible by 6 .

Now considering the above

Since $3(k+1)(k+2)$is divisible by 3 and also product of two consecutive numbers is even, it follows that $3(k+1)(k+2)$is divisible by 6 and $k(k+1)(k+2)$is divisible by 6 , it follows that $(k+1)(k+2)(k+3)$is divisible by 6 .

As a result statement is true for all natural numberswidth="11">n

$(k+1)(k+2)(k+3)=k(k+1)(k+2)+3(k+1)(k+2)$