Question 72435
{{{(1/(sqrt(1) + sqrt(2)))+(1/(sqrt(2) + sqrt(3)))+(1/(sqrt(3) + sqrt(4)))+(1/(sqrt(4) + sqrt(5)))}}}...{{{(1/(sqrt(99) + sqrt(100)))}}}


= {{{((sqrt(2) - sqrt(1))/((sqrt(2) - sqrt(1))(sqrt(2) + sqrt(1))))+
((sqrt(3) - sqrt(2))/((sqrt(3) - sqrt(2))(sqrt(3) + sqrt(2))))+
((sqrt(4) - sqrt(3))/((sqrt(4) - sqrt(3))(sqrt(4) + sqrt(3))))+
((sqrt(5) - sqrt(4))/((sqrt(5) - sqrt(4))(sqrt(5) + sqrt(4))))}}}...{{{((sqrt(100) - sqrt(99))/((sqrt(100) - sqrt(99))(sqrt(100) + sqrt(99))))}}}


= {{{((sqrt(2) - sqrt(1))/(2-1))+
((sqrt(3) - sqrt(2))/(3-2))+
((sqrt(4) - sqrt(3))/(4-3))+
((sqrt(5) - sqrt(4))/(5-4))}}}...{{{((sqrt(100) - sqrt(99))/(100 -99))}}}________[since {{{(a+b)(a-b) = a^2 - b^2}}}]


= {{{((sqrt(2) - sqrt(1))/1)+
((sqrt(3) - sqrt(2))/1)+
((sqrt(4) - sqrt(3))/1)+
((sqrt(5) - sqrt(4))/1)}}}...{{{((sqrt(100) - sqrt(99))/1)}}}


= {{{(sqrt(2) - sqrt(1))+
(sqrt(3) - sqrt(2))+
(sqrt(4) - sqrt(3))+
(sqrt(5) - sqrt(4))}}}...{{{(sqrt(100) - sqrt(99))}}}


= {{{sqrt(100) - sqrt(1)}}}


= {{{10 - 1}}}


= {{{9}}}