A₁,A_2,A₃,.....A_n are points on the line y=x lying in the positive quadrant such that OA_n=n.OA_n-1,'O' being the origin .if OA₁=1 and the coordinate s of A_n are (2520√2,2520√2),then the value of n is

To solve the problem, we need to analyze the relationship between the points A1, A2, A3, ..., An on the line y = x in the positive quadrant, given the condition that the distance from the origin O to each point An is n times the distance to the previous point An-1. Let's break this down step by step.

Understanding the Points on the Line

The points A1, A2, A3, ..., An are located on the line y = x, which means that for any point Ai, its coordinates can be represented as (x, x). The distance from the origin O (0, 0) to any point Ai is given by the formula:

OAi = √(x² + x²) = √(2x²) = x√2

Establishing the Relationship

According to the problem, we know:

• OA1 = 1
• OA2 = 2 * OA1
• OA3 = 3 * OA2
• ... and so on.

From the first point, we have:

OA1 = 1 = A1√2

Thus, the coordinates of A1 are:

A1 = (1/√2, 1/√2)

Calculating Distances for Subsequent Points

Now, let's calculate the distances for the subsequent points:

For A2:

OA2 = 2 * OA1 = 2 * 1 = 2

Thus, OA2 = 2 = A2√2, leading to:

A2 = (2/√2, 2/√2) = (√2, √2)

For A3:

OA3 = 3 * OA2 = 3 * 2 = 6

So, OA3 = 6 = A3√2, which gives:

A3 = (6/√2, 6/√2) = (3√2, 3√2)

Generalizing the Pattern

From the calculations, we can see a pattern emerging:

• OA1 = 1
• OA2 = 2
• OA3 = 6

Continuing this pattern, we can express OA_n as:

OA_n = n! (n factorial), since:

• OA1 = 1! = 1
• OA2 = 2! = 2
• OA3 = 3! = 6

Finding the Value of n

We know that the coordinates of An are given as (2520√2, 2520√2). Therefore, the distance OA_n can be calculated as:

OA_n = 2520√2√2 = 2520 * 2 = 5040

Now, we need to find n such that:

n! = 5040

Calculating Factorials

Let's calculate the factorials of integers until we reach 5040:

• 1! = 1
• 2! = 2
• 3! = 6
• 4! = 24
• 5! = 120
• 6! = 720
• 7! = 5040

From this, we can see that:

7! = 5040

Conclusion

Thus, the value of n is 7. Therefore, the answer to your question is:

n = 7