Question

Compute the sum \[\frac{1}{\sqrt{100} + \sqrt{102}} + \frac{1}{\sqrt{102} + \sqrt{104}} + \frac{1}{\sqrt{104}+\sqrt{106}} + \cdots + \frac{1}{\sqrt{9998} + \sqrt{10000}}.\]

Answer 1

45

Explanation:

I did the algebra b class

Answer 2

`(1)/( √(100)+ √(102) ) + (1)/( √(102) + √(104) )+...+ (1)/( √(9998)+ √(10000) )`
We will multiply all fractions to make a difference of squares in the denominators. So this sum will become:

`( √(102) - √(100) )/(2)+ ( √(104)- √(102) )/(2)+...+ ( √(10000)- √(9998) )/(2)`
= - √100 / 2 + √10000 / 2 = - 10 / 2 + 100 / 2 =
= - 5 + 50 = 45