Question

Use mathematical induction to prove that n(n^2+5) is divisible by 6

Answer 1

Answer:

Let P(n): n(n2 + 5) is divisible by 6, for each natural number.

Now P(l): 1 (l2 + 5) = 6, which is divisible by 6.

Hence, P(l) is true.

Let us assume that P(n) is true for some natural number n = k.

P(k): k( k2 + 5) is divisible by 6.

or K (k2+ 5) = 6m, m∈N (i)

Now, we have to prove that P(k + 1) is true.

P(K+l):(K+l)[(K+l)2 + 5]

= (K + l)[K2 + 2K+6]

= K3 + 3 K2 + 8K + 6

= (K2 + 5K) + 3 K2 + 3K + 6 =K(K2 + 5) + 3(K2 + K + 2)

= (6m) + 3(K2 + K + 2) (using (i))

Now, K2 + K + 2 is always even if A is odd or even.

So, 3(K2 + K + 2) is divisible by 6 and hence, (6m) + 3(K2 + K + 2) is divisible by 6.

Hence, P(k + 1) is true whenever P(k) is true.

So, by the principle of mathematical induction P(n) is true for any natural number n.