Σx=199 (1)/(x(x+1)) = …

Official answer: 0.99. Explanation: We can rewrite the general term using partial fractions: (1)/(x(x+1)) = (1)/(x) - (1)/(x+1) Applying this to the summation: Σx=199 (1)/(x(x+1)) = Σx=199 ( (1)/(x) - (1)/(x+1) ) This is a telescoping series: = (1 - (1)/(2) ) + ((1)/(2) - (1)/(3) ) + ((1)/(3) - (1)/(4) ) + + ((1)/(99) - (1)/(100) ) All intermediate terms cancel out: = 1 - (1)/(100) = (99)/(100) = 0.99 Thus, the answer is 0.99.

Calculus

GATE CSE 2015 Set 1 · Q26

NAT1 markEasy

Time00:00

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

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Enter a value to check

Solution

Step-by-step verified solution

Official answer: 0.99

Calculus

We can rewrite the general term using partial fractions:

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

Applying this to the summation:

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

This is a telescoping series:

= \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)

All intermediate terms cancel out:

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

Thus, the answer is 0.99.

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This Calculus previous year question is Q26 of the GATE CSE 2015 Set 1 paper. Find more GATE CSE PYQs on Calculus in Engineering Mathematics, or browse all GATE CSE PYQs.

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Calculus in GATE CSE

Calculus has 40 GATE CSE PYQs across 25 papers from 2011 to 2026. 11 of them are 1-mark NAT questions like this one, and GATE CSE 2015 Set 1 has 4.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.