(1)/(1+√(2) ) +(1)/(√(2) +√(3) ) +...+(1)/(√(99) +√(100) )

Question

`(1)/(1+√(2) ) +(1)/(√(2) +√(3) ) +...+(1)/(√(99) +√(100) )`

Answer 1

9

Explanation:

the given expression can be re-written:

`(√(2) -1)/((√(2)-1)(√(2)+1)) +(√(3) -√(2))/((√(3) +√(2))(√(3) -√(2))) + ...+(√(100) -√(99))/((√(100) -√(99))(√(100) +√(99)));`

then

`√(2) -1+√(3) -√(2) +√(4) -√(3)+√(5) -√(4)+...+√(100)-√(99)= -1+√(100)=9.`