Question:
If $a_1, a_2, a_3, a_4$ are in A.P., then $\frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \frac{1}{\sqrt{a_2} + \sqrt{a_3}} + \frac{1}{\sqrt{a_3} + \sqrt{a_4}} =$
If $a_1, a_2, a_3, a_4$ are in A.P., then $\frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \frac{1}{\sqrt{a_2} + \sqrt{a_3}} + \frac{1}{\sqrt{a_3} + \sqrt{a_4}} =$
Show Hint
In any A.P., the sum of terms equidistant from the beginning and the end is always constant. This means \( a_1 + a_n = a_2 + a_{n-1} = a_3 + a_{n-2} \), etc. This property makes finding the total sum very quick if you have any of these pairs.
Updated On: May 2, 2026
- \( 90 \)
- \( 900 \)
- \( 350 \)
- \( 450 \)
- \( 730 \)
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The Correct Option is D
Solution and Explanation
Concept:
The sum of the first \( n \) terms of an Arithmetic Progression (A.P.) can be calculated using the formula:
\[
S_n = \frac{n}{2}(a + l)
\]
where \( n \) is the number of terms, \( a \) is the first term (\( a_1 \)), and \( l \) is the last term (\( a_n \)).
Step 1: Identify the known values from the problem.
We are looking for the sum of the first 20 terms (\( S_{20} \)).
• Number of terms (\( n \)) = 20
• Sum of first and last term (\( a_1 + a_{20} \)) = 45
Step 2: Substitute values into the sum formula.
Using the formula \( S_{20} = \frac{20}{2}(a_1 + a_{20}) \): \[ S_{20} = 10 \times (45) \] \[ S_{20} = 450 \]
Step 1: Identify the known values from the problem.
We are looking for the sum of the first 20 terms (\( S_{20} \)).
• Number of terms (\( n \)) = 20
• Sum of first and last term (\( a_1 + a_{20} \)) = 45
Step 2: Substitute values into the sum formula.
Using the formula \( S_{20} = \frac{20}{2}(a_1 + a_{20}) \): \[ S_{20} = 10 \times (45) \] \[ S_{20} = 450 \]
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