Prove the following by using the principle of mathematical induction for all `n in N`:`(1+3/1)(1+5/4)(1+7/9)dotdotdot(1+((2n+1))/(n^2))=(n+1)^2`

Explanation

`" Let "P(n) : (1+3)(1+(5)/(4))(1+(7)/(9))`
`......(1+(2n+1)/(n^(2)))=(n+1)^(2)`
For n=1
`L.H.S. =1+3 =4,R.H.S. =(1+1)^(2)=4`
`:. 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=3)(1+(5)/(4)) (1+(7)/(9))`
`......(1+(2k+1)/(k^(2))) =(k+1)^(2)`
For n=K+1
`P(k+1) : (1+3) (1+(5)/(4)) (1+(5)/(9))`
`......(1+(2K+1)/(k^(2))).{1+(2(k+1)+1)/((k+1))}`
`=(k+1)^(2) {1+(2(K+1)+1)/((K+1)^(2))}`
`=(k+1)^(2){((k+1)^(2)+2(K+1)+1)/((k+1)^(2)]}`
`={(K+1)+1}^(2)=(K+2)^(2)`
`rArr` P (n) is also frue for n=K+1
Hence from the principle of the mathamatical induction P (n) is true for all natural numbers n.