Question

The sum of the series \[ \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \dots \] up to 15 terms is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Telescoping series simplify by cancellation, reducing long sums into simple expressions.

Answer 1

The Correct Option is C

Step 1: Simplifying each term.
Using rationalization, \[ \frac{1}{\sqrt{k} + \sqrt{k+1}} = \sqrt{k+1} - \sqrt{k} \] Step 2: Summing the series.
The series telescopes, leaving: \[ \sqrt{16} - \sqrt{1} = 4 - 1 = 3 \] Thus, the correct answer is (C).