Prove by PMI that `1.2+ 2.3+3.4+....+ n(n+1) =((n)(n+1)(n+2))/3, AA n in N`

Prove by PMI that `1.2+ 2.3+3.4+....+ n(n+1) =((n)(n+1)(n+2))/3, AA n

1.	Prove by PMI that `1.2+ 2.3+3.4+....+ n(n+1) =((n)(n+1)(n+2))/3, AA n in N`
Answer» Let `p (n) : 1.2 +2.3+3.4+â€¦..+n.(n+1)` `=1/3n(n+1)(n+2)` For n =1 `L.H.S. =1.2=2` `R.H.S. =1/3.1.(1+1)(1+2)=2` `:. L.H.S. =R.H.S.` `rArr` P (n) is true for n=1 Let P (n) be true for n=k. `:. P (k) :1.2+2.3+3.4+....+k.(k+1)` `=1/3 k(k +1)(k+2)` For n=K+1 `P(k +1) :1.2+2.3+3.4+....+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))/(3)` `rArr` P (n) is also true for n=(K+1) Hence from the principle of mathematical indicution P (n) is true for all natural numbers n.