Question

Prove that 1.2 + 2.3 + 3.4 +..............n(n+1) = 1/3 n (n+1)(n+2)

Answer 1

We shall prove the result by principle of mathematical induction. 

checking for n = 1, 

LHS : 1.2 = 2
RHS : 1/3 * 1 * 2 * 3 = 2.
Hence true for n = 1

Let us assume the result is true for n = k. 
that is, 1.2 + 2.3 +.....k(k+1) = 1/3 * k * (k+1) * (k+2) 

We shall prove the result to be true for n = k+1. 
that is, to prove 1.2 + 2.3 .....+ k(k+1) + (k+1)(k+2) = 1/3 (k+1) (k+2) (k+3)

consider LHS:
1.2 + 2.3 .....+ k(k+1) + (k+1)(k+2) 
= 1/3 * k * (k+1) * (k+2) + (k+1)(k+2)
=(k+1)(k+2) [1/3*k + 1]
=(k+1)(k+2)(k+3)1/3
=RHS. 

Hence the result holds for n=k+1. 

Hence proof is complete by PMI and therefore the result holds.