To solve this equation, you would first have to start by rationalizing the denominator of each term. After rationalizing the denominator, the sequence will appear as (sqrt2-sqrt1)+(sqrt3-sqrt2)+(sqrt4-sqrt3)+(sqrt(a)-sqrt(a-1))+(sqrt100-sqrt99). From here, the sequence can be quickly simplified. In the first term, sqrt2 appears and in the second term, -sqrt2 appears. this means that the sqrt2 can be eliminated from the sequence. This elimination is possible for all of the square roots except for the -sqrt1 in the first term and the sqrt100 in the last term due to there being a sqrt(a) and a corresponding -sqrt(a) for all square roots except these two. This leaves us with a simplified sequence of -sqrt1+sqrt100, which can be simplified to -1+10, which is equal to 9. Therefore, the solution to this sequence is 9.
This rule would work for any sequence with terms such as (1/(sqrt(a)+sqrt(b)))+(1/(sqrt(b)+sqrt(c)))+...+(1/(sqrt(y)+sqrt(z))). For the solution to have a whole number, -sqrt(a)+sqrt(z) would have to equal a whole number. For example, the sequence (1/(sqrt4+sqrt7))+(1/(sqrt7+sqrt10))+...+(1/(sqrt166+sqrt169)) would have a whole number solution of 11 because the sequence could eventually be simplified to -sqrt4+sqrt169, which equals 11.