Question

If \(\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \dots + \frac{1}{\sqrt{99} + \sqrt{100}} = m\) and  \(\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{99 \cdot 100} = n,\) then the point \( (m, n) \) lies on the line

• A. 11(x – 1) – 100(y – 2) = 0
• B. 11(x – 2) – 100(y – 1) = 0
• C. 11(x – 1) – 100y = 0
• D. 11x – 100y = 0

Answer 1

The Correct Option is D

Let \( m \) be the sum:

\[ \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}} = m \]

Rationalizing each term yields:

\[ \frac{\sqrt{1} - \sqrt{2}}{-1} + \frac{\sqrt{2} - \sqrt{3}}{-1} + \cdots + \frac{\sqrt{99} - \sqrt{100}}{-1} = m \]

This sum telescopes to:

\[ \sqrt{100} - \sqrt{1} = m \implies m = 10 - 1 = 9 \]

Let \( n \) be the sum:

\[ \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{99 \cdot 100} = n \]

Rewriting each term using partial fractions:

\[ \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{99} - \frac{1}{100}\right) = n \]

This sum telescopes to:

\[ 1 - \frac{1}{100} = n \implies n = \frac{99}{100} \]

Therefore, \((m, n) = (9, \frac{99}{100})\).

Substitute these values into the equation \( 11x - 100y \):

\[ 11(9) - 100\left(\frac{99}{100}\right) = 99 - 99 = 0 \]

Correct Answer: option (4) \( 11x - 100y = 0 \)