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Chapter 12: Q 22. (page 830)

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 by6

Short Answer

Expert verified

The given statement is shown.

Step by step solution

01

Step 1. Given information

The expression isn(n+1)(n+2)

02

Step 2. Show that the given statement is correct.

First we consider the statement for n=1

Because 1(1+1)(1+2)=6is 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)

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