If the line $\sqrt{5} = y$ meets the lines $x = 1., x = 2, . . ., x = n$ at points $A_{1}, A_{2}, . . . . . . A_{n}$ respectively, then $(O A_{1})^{2} + (O A_{2})^{2} . . . . + {(O A_{n})}_{n}^{2} =$

3 n^{2} + 3 n

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2 n^{2} + 4 n

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3 n^{3} + 3 n^{3} + 2

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5 n + \frac{n (n + 1) (2 n + 1)}{6}

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Solution

The correct option is D $5 n + \frac{n (n + 1) (2 n + 1)}{6}$
$O A_{1} = \sqrt{6}$ $\Rightarrow O A_{1}^{2} = 6$
$O A_{2} = \sqrt{9}$ $\Rightarrow O A_{2}^{2} = 9$
$O A_{3} = \sqrt{14}$ $\Rightarrow O A_{3}^{2} = 14$
$∴ O A_{1}^{2} + O A_{2}^{2} + O A_{3}^{2} +$ .......... $O A_{n}^{2}$
$S_{n} = 6 + 9 + 14 + . . . + (n^{2} + 5)$
$T_{n} = (n^{2} + 5)$
$S_{n} = n \sum n = 1 T n = n \sum n = 1 (n^{2} + 5)$
$S_{n} = 5 n + n \sum n = 1 n^{2}$
$S_{n} = 5 n + \frac{n (n + 1) (2 n + 1)}{6}$

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If the line √5=y meets the lines x=1.,x=2,...,x=n at points A1,A2,......An respectively, then (OA1)2+(OA2)2....+(OAn)2=

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If the line $\sqrt{5} = y$ meets the lines $x = 1., x = 2, . . ., x = n$ at points $A_{1}, A_{2}, . . . . . . A_{n}$ respectively, then $(O A_{1})^{2} + (O A_{2})^{2} . . . . + {(O A_{n})}_{n}^{2} =$