Let `S=(sqrt(1))/(1+sqrt1+sqrt(2))+sqrt(2)/(1+sqrt(2)+sqrt(3))+(sqrt(3))/(1+sqrt(3)+sqrt(4))+...+(sqrt(n))/(1+sqrt(n)+(sqrtn+1))=10`
Then find the value of n.

Answer

n=24

Explanation

`T_(r)=(sqrtr)/(1+sqrtr+sqrt(r+1))=(sqrtr{1+sqrtr-sqrt(r+1)})/(1+r+2sqrtr-(r+1))`
`=1/2{1+sqrtr-sqrt(r+1)}`
`thereforeS_(n)=1/2(n+1)-sqrt(n+1)=10`
Let `sqrt(n+1)`=x
`thereforex^(2)-x=20`
`rArrx^(2)-x-20=0`
`rArrx=sqrt(n+1)=5`
`thereforen=24`