First slide

Arithmetic progression

Question

$Let a_{1}, a_{2}, a_{3}, \dots a_{49}, a_{50} are in A.P. If a_{1} = 4, and a_{50} = 144 then the value of$ $\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{1}{\sqrt{a_{2}} + \sqrt{a_{3}}} + .. + \frac{1}{\sqrt{a_{49}} + \sqrt{a_{50}}} =$

Moderate

A

$\frac{14}{13}$
B

$\frac{49}{14}$
C

$\frac{50}{13}$
D

$\frac{48}{17}$

Solution

$\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{1}{\sqrt{a_{2}} + \sqrt{a_{3}}} + .. + \frac{1}{\sqrt{a_{49}} + \sqrt{a_{50}}} = \frac{(\sqrt{a_{2}} - \sqrt{a_{1}}) + (\sqrt{a_{3}} - \sqrt{a_{2}}) + .. + (\sqrt{a_{50}} - \sqrt{a_{49}})}{d}$
$= \frac{\sqrt{a_{50}} - \sqrt{a_{1}}}{d} = \frac{\sqrt{144} - \sqrt{4}}{\frac{140}{49}} = \frac{49}{14}$

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