Question

use the principle of mathematical induction to prove that 1.2+2.3+3.4+....+n(n+1)=1/3.n(n+1)(n+2) n belongs to N.

Answer 1

1.2+2.3+3.4+.....+n(n+1)=1/3(n(n+1)(n+2))
L.H.S
if n=1 then L.H.S=2
THEN R.H.S=2
if n=k then we can prove that
1.2+2.3+3.4+.....+k(k+1)=1/3(k(k+1)(k+2))
let's prove it is true for n=k+1 we add (k+1)(k+2) on both sides
1.2+2.3+3.4+.....+k(k+1)+ (k+1)(k+2)=1/3(k(k+1)(k+2))+(k+1)(k+2)
1.2+2.3+3.4+.....+k(k+1)+ (k+1)(k+2)=1/3(k+1)(k+2(k+3)
By the mathematical induction n belongs to N