Using the principle of mathematical induction, prove that `:` `1. 2. 3+2. 3. 4++n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4^` for all `n in N` .

Explanation

Let P (n) `:1.2.3+2.3.4+3.4.5+…..` ,
`+n(n+1)(n+2)=1/4n(n+1)(n+2)(n+3)`
For n=1
`L.H.S. =1.2.3=6`
`R.H.S. =1/4 .1.(1+1)(1+2)(1+3)=6`
`:. " "L.H.S. =R.H.S.`
`rArr` P (n) is true for n=k.
`:. P (k) : 1.2.3+2.3.4+3.4.5+`
`.......+K(K+1)(k+2)`
`=1/4 k(k+1)(k+2)(k+3)`
For n =k+1
`p(k+1) :1.2.3+2.3.4+3.4.5+......+k(K=1)`
`(k+2)+(k+1)(K+2)(K+3)`
`=1/4 k(k+1)(k+2)(k+3)+(k+1)(K+2)(K+3)` ,
`=(k+1)(K+2)(K+3)((K)/(4)+1)`
`=1/4 (k+1)(K+2)(K+3)(K+4)`
`rArr` P (n) si also true for n=K+1
Hence from the principle of mathematical induction P(n) is true for all natural numbers n.