Question

prove that n(n+1)(n+5) is a multiple of 6

Answer 1

Use principle of induction 
Hypothesis n(n+1)(n+2) is a multiple of 6 
when n=1 
n(n+1)(n+2) =1(1+1)(1+2) =6 hence true for n=1 
Assume it is true for n=k 
i.e k(k+1)(k+2) is a multiple of 
To prove that it is also true for n= k+1 if we assume it is true for n=k 
For n=k+1 
n(n+1)(n+2) =(k+1)(k+1+1)(k+1+2) 
=(k+1)(k+2)(k+3) = (k+1)(k+2)(k) +(k+1)(k+2)(3) 
=Multiple of 6 + (k+1)(k+2)(3) 
For (k+1)(k+2)(3) if k is odd k+1 is even 
Hence (k+1)(k+2)(3)is a multiple of 6 
Whenk is eve k+2 is even (k+1)(k+2)(3) is multiple of 6 
Hence n(n+1)(n+2) is a multiple of 6 for all values of n Є N

Hope it helps you...