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Q.

$A_{1}, A_{2} \dots A_{n}$ are points on the line $y = x$ lying in the positive quadrant such that $O A_{n} = n O A_{n - 1}, O$ being the origin. If $O A_{1} = 1$ and the coordinates of $A_{n}$ are $(2520 \sqrt{2}, 2520 \sqrt{2})$ , then $n =$

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a

7

b

8

c

6

d

5

answer is C.

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Detailed Solution

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We have

$\begin{array}{l} O A_{n} = n . O A_{n - 1} = n (n - 1) O A_{n - 2} \dots \\ = n (n - 1) \dots 2 (O A_{1}) = n! O A_{1} = n! \\ \Rightarrow \sqrt{2} (2520 \sqrt{2}) = n! \\ \Rightarrow 2520 \times 2 = n! \\ \Rightarrow 7 \times 6 \times 5 \times 4 \times 3 \times 2 = n! \\ \Rightarrow n = 7 \end{array}$

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