Question

prove by mathematical induction that n(n+ 1) (2n+ 1) is divisible by 6 if n s a natural number​

Answer 1

1. Let n = 1.

n(n + 1)(2n + 1)

1(1 + 1)(2 × 1 + 1)

1 × 2 × 3 = 6

We got 6.

∴ n(n + 1)(2n + 1) is divisible by 6 if n = 1.

2. Let n = k.

Assume that k(k + 1)(2k + 1) is divisible by 6.

3. Let n = k + 1

n(n + 1)(2n + 1)

(k + 1)(k + 1 + 1)(2(k + 1) + 1)

(k + 1)(k + 2)(2k + 2 + 1)

(k + 1)(k + 2)(2k + 1 + 2)

(k + 1)(k + 2)(2k + 1) + (k + 1)(k + 2)2

k(k + 1)(2k + 1) + 2(k + 1)(2k + 1) + (k + 1)(k + 2)2

k(k + 1)(2k + 1) + 2(k + 1)(2k + 1 + k + 2)

k(k + 1)(2k + 1) + 2(k + 1)(3k + 3)

k(k + 1)(2k + 1) + 2(k + 1)(k + 1)3

k(k + 1)(2k + 1) + 6(k + 1)²

Here, we assumed earlier that k(k + 1)(2k + 1) is a multiple of 6 at the second step. To this, 6(k + 1)² is added which is also a multiple of 6 as (k + 1)² multiplied by 6 is added to k(k + 1)(2k + 1).

∴ n(n + 1)(2n + 1) is divisible by 6 if n = k + 1.

∴ n(n + 1)(2n + 1) is divisible by 6 if n is a natural number.

Hence proved!!!

This answer is on my own words. Trust me. Not from any website or any books or from any other source.

Please ask me any doubts if you're confused on this.

Thank you. Have a nice day. :-)

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