Find the value of - a - - a - n d - a - - a - d - - a - - a - n d - a - d - - a - 2 d - - - - a - - a - n d - a - n - 1 d - - a - n d

The correct option is D n Given #160; √(a)+√(a+nd)/√(a)+√(a+d)+√(a)+√(a+nd)/√(a+d)+√(a+2d)+...+√(a)+√(a+nd)/√(a+(n 1)d)+√(a+nd) Raltionalize the deno ...

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Byju's Answer

Standard X

Mathematics

Sum of N Terms of an AP

Find the valu...

Question

Find the value of $\frac{\sqrt{a} + \sqrt{a + n d}}{\sqrt{a} + \sqrt{a + d}} + \frac{\sqrt{a} + \sqrt{a + n d}}{\sqrt{a + d} + \sqrt{a + 2 d}} + \dots + \frac{\sqrt{a} + \sqrt{a + n d}}{\sqrt{a + (n - 1) d} + \sqrt{a + n d}}$

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d

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n

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(n - 1) d

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Solution

The correct option is D $n$
Given $\frac{\sqrt{a} + \sqrt{a + n d}}{\sqrt{a} + \sqrt{a + d}} + \frac{\sqrt{a} + \sqrt{a + n d}}{\sqrt{a + d} + \sqrt{a + 2 d}} + . . . + \frac{\sqrt{a} + \sqrt{a + n d}}{\sqrt{a + (n - 1) d} + \sqrt{a + n d}}$

Raltionalize the denominators

$= (\sqrt{a} + \sqrt{a + n d}) (\frac{1}{\sqrt{a} + \sqrt{a + d}} + \frac{1}{\sqrt{a + d} + \sqrt{a + 2 d}} + . . . + \frac{1}{\sqrt{a} + (n - 1) d + \sqrt{a + 2 d}})$

$= (\sqrt{a} + \sqrt{a + n d}) (\frac{1}{\sqrt{a} + \sqrt{a + d}} \times \frac{\sqrt{a + d} - \sqrt{a}}{\sqrt{a + d} - \sqrt{a}} + \frac{1}{\sqrt{a + d} + \sqrt{a + 2 d}} \times \frac{\sqrt{a + 2 d} - \sqrt{a + d}}{\sqrt{a + 2 d} - \sqrt{a + d}}$

$+ . . . + \frac{1}{\sqrt{a + (n - 1) d + \sqrt{a + n d}}} \times \frac{\sqrt{a + n d} - \sqrt{a + (n - 1) d}}{\sqrt{a + n d} - \sqrt{a + (n - 1) d}})$

$= (\sqrt{a} + \sqrt{a + n d}) (\frac{\sqrt{a + d} - \sqrt{a}}{a + d - a} + \frac{\sqrt{a + 2 d} - \sqrt{a + d}}{a + 2 d - (a + d)} + \frac{\sqrt{a + 3 d} - \sqrt{a + 2 d}}{a + 3 d - (a + 2 d)}$

$+ . . . + \frac{\sqrt{a + n d} - \sqrt{a + (n - 1) d}}{a + n d - (a + (n - 1) d)})$

$= (\sqrt{a} + \sqrt{a + n d}) (\frac{\sqrt{a + d} - \sqrt{a}}{d} + \frac{\sqrt{a + 2 d} - \sqrt{a + d}}{d} + \frac{\sqrt{a + 3 d} - \sqrt{a + 2 d}}{d}$

$+ . . . + \frac{\sqrt{a + n d} - \sqrt{a + (n - 1) d}}{d})$

$= (\sqrt{a} + \sqrt{a + n d}) (\frac{\sqrt{a + d} - \sqrt{a} + \sqrt{a + 2 d} - \sqrt{a + d} + \sqrt{a + 3 d} - \sqrt{a + 2 d}}{d}$

$+ . . . . + \frac{\sqrt{a + n d} - \sqrt{a + (n - 1) d}}{d})$

$= \frac{(\sqrt{a} + \sqrt{a + n d}) (\sqrt{a + n d} - \sqrt{a})}{d}$

$= \frac{(a + n d - a)}{d} = n$

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Find the value of √a+√a+nd√a+√a+d+√a+√a+nd√a+d+√a+2d+…+√a+√a+nd√a+(n−1)d+√a+nd

Find the value of $\frac{\sqrt{a} + \sqrt{a + n d}}{\sqrt{a} + \sqrt{a + d}} + \frac{\sqrt{a} + \sqrt{a + n d}}{\sqrt{a + d} + \sqrt{a + 2 d}} + \dots + \frac{\sqrt{a} + \sqrt{a + n d}}{\sqrt{a + (n - 1) d} + \sqrt{a + n d}}$