Prove the following by the principle of mathematical induction: `1/(3. 5)+1/(5. 7)+1/(7. 9)+1/((2n+1)(2n+3))=n/(3(2n+3))`

Explanation

`" Let " P(n) : (1)/(3.5) +(1)/(5.7)+(1)/(7.9)+......`
` +(1)/((2n+1)(2n+3))=(n)/(3(2n+3))`
For n=1 , L.H.S. `=(1)/(3.5)`
`R.H.S. =(1)/(3(2.1+3))=(1)/(3(2+3))+(1)/(3.5)`
`:. " "L.H.S. =R.H.S`
Therefore given statement P (n) is true for n=1
Let P (n) be true for n=k
`:. P(k) : (1)/(3.5) +(1)/(5.7) +(1)/(7.9)+......`
`+(1)/((2k+1)(2k+3))=(k)/(3(2k+3))`
For n=K+1
`P(k+1):(1)/(3.5)+(1)/(5.7)+(1)/(7.9)+......+(1)/((2k+1)(2K+3))`
`(1)/((2k+3)(2K+5))`
`=((1)/(3.5)+(1)/(5.7)+(1)/(7.9) +.....+(1)/((2k+1)(2k+3)))`
`+(1)/((2k+3)(2K+5))`
`=(k)/(3(2k+3))+(1)/((2K+3)(2k+5))`
`=(1)/(3(2k+3))[k+(3)/(2k+5)]`
`=(1)/(3(2k+3)).(2k^(2)+5k+3)/((2k+5))`
`=(k+1)/(3(2K+5))`
P (n) is also true for n=K+1
Hence from the principle of mathematical induction P (n) is true for all natural numbers n.