Question

show that one and only one out of n, n(n+1),(n+2) is divisible by 3, when n is any positive integer

Answer 1

MATHEMATICAL INDUCTION

1. Let n = 1.

=> n(n+1)(n+2)

=> 1 × 2 × 3

=> 3

3 is divisible by 3.

So n(n+1)(n+2) is divisible by 3 if n = 1.

2. Let n = k.

Assume that k(k+1)(k+2) is divisible by 3.

3. Let n = k+1.

=> n(n+1)(n+2)

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

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

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

Here, we assumed earlier that k(k+1)(k+2) is a multiple of 3. To this, 3(k+1)(k+2), which is also a multiple of 3, is added.

Therefore, n(n+1)(n+2) is divisible by 3 in this instance.

Therefore, n(n+1)(n+2) is divisible by 3 for all positive integers n.

Hence proved!!!

Thank you. :-))

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