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

Explanation

Let
`p (n) : 1.2 +2.3+3.4+…..+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.