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

3 n^{2} + 3 n

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

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

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(\frac{3}{2}) (n^{4} + 2 n^{3} + n^{2})

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Solution

The correct option is B $2 n^{3} + 3 n^{2} + n$
Let $A_{k} (k, \sqrt{5} k)$ be the general point where the given line meets $x = k$
The distance of this point from the origin is $O A_{k} = \sqrt{(k - 0)^{2} + (\sqrt{5} k - 0)^{2}} = \sqrt{6} k$ .
Then, $O A_{1}^{2} + O A_{2}^{2} + . . . . . . . . . . . . . . . . . . . O A_{n}^{2} = 6 (1^{2} + 2^{2} + . . . . . . . . n^{2})$
= $2 n^{3} + 3 n^{2} + n$

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