Prove by pmi that n(n+1)(2n+1) is divisible by 6 for all natural numbe

1.	Prove by pmi that n(n+1)(2n+1) is divisible by 6 for all natural numbers
Answer» 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 + 2) is divisible by 6 for all n N.

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