Prove by induction that 3n(n 1) is divisible by 6 for all positive integers.

We are asked to prove by the method of mathematical induction that:

      3n(n+1) is divisible by 6 for all positive integers.

which is divisible by 6.

Hence, the result is true for n=1

i.e. 3k(k+1) is divisible by 6.

Let n=k+1

then

Since, the first term:

is divisible by 6.

( As the result is true for n=k)

and the second term is also divisible by 6.

Hence, the sum:

 is divisible by 6.

Hence, the result is true for n=k+1

Hence, we may say that the result is true for all n where n belongs to positive integers.

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