GATE CS 2015 Set 1 Q26 — Combinatorics: Counting, Recurrence Relations, Generating Functions

The given summation is a classic telescoping series:

$$\sum_{x=1}^{99} \frac{1}{x(x+1)}$$

Using partial fraction decomposition, we can rewrite the general term as:

$$\frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1}$$

Substituting this back into the summation, we get:

$$\sum_{x=1}^{99} \left( \frac{1}{x} - \frac{1}{x+1} \right)$$

Expanding the terms of the sum:

$$S = \left( 1 - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \dots + \left( \frac{1}{99} - \frac{1}{100} \right)$$

Notice that all intermediate terms cancel out (telescoping effect):

$$S = 1 - \frac{1}{100} = \frac{99}{100} = 0.99$$

Thus, the value of the summation is 0.99.