Question

Prove that n(n+1)(n+2) is divisible by 6​

Answer 1

Answer:

We also have either one of n or n+1 is divisible by 2 because they are two consecutive natural numbers. Thus, the product of n, n+1 and n+2 is always divisible by 2∗3=6. If a number is divisible by 6=2⋅3, then it must be divisible by 2, and also by 3.

Step-by-step explanation:

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Answer 2

Step-by-step explanation:

Let P(n): n(n + 1)(n + 2) is divisible by 6.

P(1): 1(1 + 1)(1 + 2) = 6 which is divisible by 6. Thus P(n) is true for n = 1.

Let P(k) be true for some natural number k.

i.e. P(k): k(k + 1)(k + 2) is divisible by 6.

Now we prove that P(k + 1) is true whenever P(k) is true.

Now, (k+1)(k+2)(k+3) = k(k+1)(k+2)+3(k+1)(k+2)

Since, we have assumed that k(k+1)(k+2) is divisible by 6 , also (k+1)(k+2) is divisible by 6 as either of (k+1) and (k+2) has to be even number.

P(k+1) is true.

Thus P(k+1) is true whenever P(k) is true.

By Principle mathematical induction n(n+1)(n+3) is divisible by 6.