Question

the value of $\frac{1}{ \sqrt{2} + 1 } + \frac{1}{ \sqrt{3}+ \sqrt{2} } + \frac{1}{ \sqrt{4}+ \sqrt{3} } +...........+ \frac{1}{ \sqrt{9}+ \sqrt{8} }$ is?

Answer 1

1/(√2 +1) + 1/(√3 + √2) + 1/(√4 +√3) + ........1/(√9 + √8) = ??

use concept of rationalisation ,
1/(√2 + 1) = √2 -1
1/(√3 +√2) = √3 -√2
1/(√4+√3) = √4 -√3
1/(√5 + √4) = √5 -√4
.. ..,.....
................
1/(√9 +√8) = √9 -√8

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now add all of these ,
you can see that,√2 of (√2 -1) subtract by √2 of (√3 -√2) , in this way 1st term of first subtract by second of second ,
and then come final result = √9 -1

hence, answer is √9 -1