Use the method of induction to prove that n(n+1)(n+2)is a multiple of 6 for every n belongs to N - Maths - Principle of Mathematical Induction

Question

Use the method of induction to prove that n(n+1)(n+2)is a multiple
of 6 for every n belongs to N

Manbar Singh · Accepted Answer

Let pn: nn+1n+2 is a multiple of 6 For n = 1, p1 = 11+11+2 = 6 ,
which is a multiple of 6 So, p1 is true Let pk be true Then,pk :
kk+1k+2 is a multiple of 6 ⇒kk+1k+2 = 6m for some natural
number m Now, we have to prove, pk+1 is true whenever pk is true We
have,pk+1 : k+1k+2k+3now,k+1k+2k+3 = k+3k+1k+2 = kk+1k+2 + 3k+1k+2
= 6m + 3k+1k+2now, k+1 and k+2 are consecutive integers, so their
product is even then, k+1 k+2= 2q evenk+1k+2k+3 = 6m + 3 2q = 6m +
q, which is a multiple of 6 ⇒pk+1 : k+1k+2k+3 is a multiple
of 6 ⇒ pk+1 is true whenever pk is true Thus, p1 is true and
pk+1 is true, whenever pk is true Hence by the principal of
mathematical induction, pn is true for all n∈ N