Question

How do you simplify `1/(2+sqrt(5))+1/(sqrt(5)+sqrt(6))+1/(sqrt(6)+sqrt(7))+1/(sqrt(7)+sqrt(8))` ?

Answer 1

`2sqrt(2)-2`

Explanation:

Note that:

`1/(sqrt(n)+sqrt(n+1)) = (sqrt(n+1)-sqrt(n))/((sqrt(n+1)-sqrt(n))(sqrt(n+1)+sqrt(n))`

`color(white)(1/(sqrt(n)+sqrt(n+1))) = (sqrt(n+1)-sqrt(n))/((n+1)-n)`

`color(white)(1/(sqrt(n)+sqrt(n+1))) = sqrt(n+1)-sqrt(n)`

So:

`1/(2+sqrt(5))+1/(sqrt(5)+sqrt(6))+1/(sqrt(6)+sqrt(7))+1/(sqrt(7)+sqrt(8))`

`=(color(red)(cancel(color(black)(sqrt(5))))-sqrt(4))+(color(orange)(cancel(color(black)(sqrt(6))))-color(red)(cancel(color(black)(sqrt(5)))))+(color(green)(cancel(color(black)(sqrt(7))))-color(orange)(cancel(color(black)(sqrt(6)))))+(sqrt(8)-color(green)(cancel(color(black)(sqrt(7)))))`

`=sqrt(8)-sqrt(4) = 2sqrt(2)-2`