Question

If $a_1, a_2, a_3, a_4$ are in A.P., then $\frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \frac{1}{\sqrt{a_2} + \sqrt{a_3}} + \frac{1}{\sqrt{a_3} + \sqrt{a_4}} =$

• A. \( \frac{\sqrt{a_4} - \sqrt{a_1}}{a_3 - a_2} \)
• B. \( \frac{a_4 - a_1}{a_3 - a_2} \)
• C. \( \frac{a_3 - a_2}{\sqrt{a_4} - \sqrt{a_1}} \)
• D. \( \frac{a_1 - a_4}{a_3 - a_1} \)
• E. \( \frac{a_5 - a_0}{a_1 - a_4} \)

Hint:

This type of sum is a "telescoping series." When you rationalize the denominators of terms involving consecutive A.P. elements, most terms in the middle will cancel out, leaving only parts of the first and last terms.

Answer 1

The Correct Option is A

Concept: In an Arithmetic Progression (A.P.), the difference between consecutive terms is constant. Let this common difference be \( d \). Therefore, \( a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = d \). To simplify expressions involving square roots in the denominator, we use the method of rationalization.

Step 1: Rationalize each term in the expression.
Consider the first term: \[ \frac{1}{\sqrt{a_1} + \sqrt{a_2}} = \frac{1}{\sqrt{a_2} + \sqrt{a_1}} \times \frac{\sqrt{a_2} - \sqrt{a_1}}{\sqrt{a_2} - \sqrt{a_1}} = \frac{\sqrt{a_2} - \sqrt{a_1}}{a_2 - a_1} = \frac{\sqrt{a_2} - \sqrt{a_1}}{d} \]

Step 2: Apply the same logic to the remaining terms.
For the second term: \[ \frac{1}{\sqrt{a_2} + \sqrt{a_3}} = \frac{\sqrt{a_3} - \sqrt{a_2}}{a_3 - a_2} = \frac{\sqrt{a_3} - \sqrt{a_2}}{d} \] For the third term: \[ \frac{1}{\sqrt{a_3} + \sqrt{a_4}} = \frac{\sqrt{a_4} - \sqrt{a_3}}{a_4 - a_3} = \frac{\sqrt{a_4} - \sqrt{a_3}}{d} \]

Step 3: Sum the rationalized terms and simplify.
Adding them together: \[ S = \frac{\sqrt{a_2} - \sqrt{a_1}}{d} + \frac{\sqrt{a_3} - \sqrt{a_2}}{d} + \frac{\sqrt{a_4} - \sqrt{a_3}}{d} \] \[ S = \frac{(\sqrt{a_2} - \sqrt{a_1}) + (\sqrt{a_3} - \sqrt{a_2}) + (\sqrt{a_4} - \sqrt{a_3})}{d} \] Notice that \( \sqrt{a_2} \) and \( \sqrt{a_3} \) cancel out: \[ S = \frac{\sqrt{a_4} - \sqrt{a_1}}{d} \]

Step 4: Match with the given options.
Since \( d = a_2 - a_1 = a_3 - a_2 \), we can substitute \( d \) in the denominator: \[ S = \frac{\sqrt{a_4} - \sqrt{a_1}}{a_3 - a_2} \] This matches option (A).