Using the principle of mathematical induction prove that `1/(1. 2. 3)+1/(2. 3. 4)+1/(3. 4. 5)++1/(n(n+1)(n+2))=(n(n+3))/(4(n+1)(n+2)` for all `n in N`

Explanation

Let `P (n) : (1)/(1.2.3)+(1)/(2.3.4)+(1)/(3.4.5)+`
`…….+(1)/(n(n+1)(n+2)) =(n(n+3))/(4(n+1)(n+2))`
For n=1
`L.H.S.=(1)/(1.2.3)=(1)/(6)`
`R.H.S. =(1.(1+3))/(4.(1+1)(1+2))=1/6`
`:. " "L.H.S. =R.H.S.`
`rArr` P (n) true for n=1
Let P(n) be true for n=K .
`P (K) : (1)/(1.2.3) +(1)/(2.3.4)+(1)/(3.4.5) +`
`.....+(1)/(K(K+1)(K+2))=(k(k+3))/((4K+1)(K+2))`
For n=K+1
`P(k+1): (1)/(1.2.3)+(1)/(2.3.4)+(1)/(3.4.5)+`
`......+(1)/(K(K+1)(K+2))+(1)/((K+1)(K+2)(K+3))`
`=(k(K+3))/((4K+1)(K+2))+(1)/((K+1)(K+2)(K+3))`
`=(k(k+3)^(2)+4)/(4(k+1)(K+2)(K+3))`
`=(k^(3)+6K^(2)+9K+4)/(4(K+1)(K+2)(K+3))`
`=((k+1)(k^(2)+5K+4))/(4(K+1)(K+2)(K+3))`
`=((K+1)(K+4))/(4(K+2)(K+3))`
`rArr` P (n) is also true for n=K +1
Hence form the the principle of mathematical induction P (n) is true for all natural numbers n.