Irrational Denominators

$\frac{1}{ 1 + \sqrt{2} } + \frac{1}{ \sqrt{2} + \sqrt{3} } + \frac{1}{ \sqrt{3} + \sqrt{4} } + .......... + \frac{1}{ \sqrt{8} + \sqrt{9} }$

$= \frac{ \sqrt{2} - 1 }{ ( \sqrt{2} )^{2} - 1 } + \frac{ \sqrt{3} - \sqrt{2} }{ ( \sqrt{3} )^{2} - ( \sqrt{2} )^{2} } + ........ + \frac{ \sqrt{9} - \sqrt{8} }{ ( \sqrt{9} )^{2} - ( \sqrt{8} )^{2} }$

$= \frac{ \sqrt{2} - 1 + \sqrt{3} - \sqrt{2} + \sqrt{4} - \sqrt{3} + ......... + \sqrt{9} - \sqrt{8} }{1}$

$= \sqrt{9} - 1 = 3 - 1 = \boxed{2}$

You can make it as a summation (like the one below):

sum(n)(start=1, end=8) [1/(sqrt(n)+sqrt(n+1))]

=sum(n)(start=1, end=8) [1/(sqrt(n)+sqrt(n+1))]*[(sqrt(n)-sqrt(n+1))/(sqrt(n)-sqrt(n+1))]

=sum(n)(start=1, end=8) (sqrt(n)-sqrt(n+1))/(n-(n+1))

=sum(n)(start=1, end=8) [ -(sqrt(n)-sqrt(n+1))]

=sum(n)(start=1, end=8) [-sqrt(n)+sqrt(n+1)]

=-1+sqrt(2)-sqrt(2)+sqrt(3)-sqrt(3)+sqrt(4)...-sqrt(8)+sqrt(9)

=-1+sqrt(9)

=-1+3

=-2 (final answer)