Calculate the value of the sum 1/(sqrt1+sqrt2) + 1/(sqrt2+sqrt3) + 1/(sqrt3+sqrt4) + ...+1/(sqrt99+sqrt100)

Because we're not allowed to keep square roots at
denominator, we’ll eliminate the them from the denominator, in this manner, by
multiplying and dividing with the conjugated
expression:

{1/[sqrt n + sqrt (n+1)]}*{[sqrt(n+1)-sqrt n]/
[sqrt(n+1)-sqrt n]}

After simplifying like
terms:

1/[sqrt n + sqrt (n+1)]=[sqrt(n+1)-sqrt
n]

So that,

S=1/ (sqrt 1+sqrt
2)]+…+[1/(sqrt99 + sqrt
100)]

S=sqrt2-sqrt1+sqrt3-sqrt2+sqrt4-sqrt3+…+sqrt100-sqrt99

S=sqrt100-sqrt1=10-1=9

S=9