Use mathematical induction to show that (1∙2) + (2∙3) + (3∙4) +⋯+ n(n+1)= [n(n+1)(n+2)]/3

Answer 1

Base case:

n = 1: 1*2 = 2 and (1*2*3)/3 = 2

Hypothesis:

(1*2)+(2*3)+ ... + (n)(n+1) = n(n+1)(n+2) / 3 for n=k

Assume n = k+1

Sum = S = (1*2)+(2*3) + ... + (k)(k+1) + (k+1)(k+2)

If we exclude the last term (k+1)(k+2), we can apply the hypothesis to the other terms and re-write S as:

S = k(k+1)(k+2) / 3 + (k+1)(k+2)

= (1/3) [ k(k+1)(k+2) + 3(k+1)(k+2) ]

= (1/3) [ (k+1)(k+2)(k+3) ]

This concludes the proof.