Question

What is the value of the sum \(\dfrac{1}{1\cdot2} + \dfrac{1}{2\cdot3} + \dfrac{1}{3\cdot4} + \cdots + \dfrac{1}{9\cdot10}\)?

• A. 9/10
• B. 1/10
• C. 10/11
• D. 8/9

Answer 1

The Correct Option is A

Step 1: Each term telescopes: \(\dfrac{1}{k(k+1)} = \dfrac{1}{k} - \dfrac{1}{k+1}\).
Step 2: Sum \(= 1 - \dfrac{1}{10} = \dfrac{9}{10}\).

Quick Tip: Split \(\dfrac{1}{k(k+1)}\) into partial fractions so consecutive terms cancel.