If `A_(1),A_(2),A_(3)` denote respectively the areas of an inscribed polygon of 2n sides , inscribed polygon of n sides and circumscribed poylgon of n sides ,then `A_(1),A_(2),A_(3)` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Answer

To solve the problem, we need to analyze the areas of the inscribed and circumscribed polygons and establish their relationships. Let's denote:

- \( A_1 \): Area of the inscribed polygon with \( 2n \) sides.
- \( A_2 \): Area of the inscribed polygon with \( n \) sides.
- \( A_3 \): Area of the circumscribed polygon with \( n \) sides.

### Step 1: Write the formula for \( A_2 \)

The area \( A_2 \) of an inscribed polygon with \( n \) sides can be given by the formula:
\[ A_2 = \frac{n a^2}{2 \sin\left(\frac{2\pi}{n}\right)} \]
where \( a \) is the length of the side of the polygon.

### Step 2: Write the formula for \( A_1 \)

For the inscribed polygon with \( 2n \) sides, the area \( A_1 \) can be expressed as:
\[ A_1 = \frac{2n a^2}{2 \sin\left(\frac{\pi}{n}\right)} = \frac{n a^2}{\sin\left(\frac{\pi}{n}\right)} \]

### Step 3: Write the formula for \( A_3 \)

The area \( A_3 \) of the circumscribed polygon with \( n \) sides can be expressed as:
\[ A_3 = n a^2 \tan\left(\frac{\pi}{n}\right) \]

### Step 4: Establish relationships between \( A_1, A_2, \) and \( A_3 \)

Now, we will relate \( A_1, A_2, \) and \( A_3 \) using the formulas we derived.

From the formula for \( A_2 \):
\[ A_2 = \frac{n a^2}{2 \sin\left(\frac{2\pi}{n}\right)} \]

Using the double angle identity for sine, we have:
\[ \sin\left(\frac{2\pi}{n}\right) = 2 \sin\left(\frac{\pi}{n}\right) \cos\left(\frac{\pi}{n}\right) \]
Thus,
\[ A_2 = \frac{n a^2}{2 \cdot 2 \sin\left(\frac{\pi}{n}\right) \cos\left(\frac{\pi}{n}\right)} = \frac{n a^2}{4 \sin\left(\frac{\pi}{n}\right) \cos\left(\frac{\pi}{n}\right)} \]

### Step 5: Relate \( A_1 \) and \( A_3 \)

Using the definitions of \( A_1 \) and \( A_3 \):
\[ A_1 = \frac{n a^2}{\sin\left(\frac{\pi}{n}\right)} \]
\[ A_3 = n a^2 \tan\left(\frac{\pi}{n}\right) = n a^2 \frac{\sin\left(\frac{\pi}{n}\right)}{\cos\left(\frac{\pi}{n}\right)} \]

### Step 6: Establish the relationship

From the relationships we derived:
\[ A_1 = 2A_2 \cos\left(\frac{\pi}{n}\right) \]
\[ A_3 = A_2 \cdot \frac{2 \sin\left(\frac{\pi}{n}\right)}{\cos\left(\frac{\pi}{n}\right)} \]

This means:
\[ A_1^2 = A_2 \cdot A_3 \]

### Conclusion

Since \( A_1^2 = A_2 \cdot A_3 \), we conclude that \( A_1, A_2, A_3 \) are in geometric progression (GP).

### Final Answer
Thus, the correct option is that \( A_1, A_2, A_3 \) are in GP.

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