Question

prove thet the product of any three consecutive positive integers is divisible by 6 solution let three consecutive numbers are n,n + 1 , + 2 ? answer.0​

Answer 1

Hence, it is proved.

Step-by-step explanation:

Let P(n) : n(n + 1)(n + 2) is a multipe of 3.

Put  n = 1, we get

P(1) : 1·(1 + 1)(1 + 2) = 1 × 2 × 3 = 6, which is divisibe by 6.

So, the given statement is true for n = 1, i.e., P(1) is true.

Let P(k) be true. Then,

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

⇒ k(k + 1)(k + 2) = 6m for some natural number m.

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

= 6[m + (k + 1)(k + 2)}

P(k + 1) : (k + 1)(k + 2)(k + 3) is divisible by 6.

⇒ p(k + 1) is true, whenever P(k) is true.