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This problem concerns regular polygons \( P_n \), where \( P_3 \) is defined as an equilateral triangle with an area of 1. The goal is to find expressions and limits related to these polygons.

### Problem breakdown:

1. \( P_3 \) is a regular triangle with an area of 1, and for \( n = 4, 5, \dots \), the area of the regular polygon \( P_n \) is denoted as \( a_n \).
   
2. The problem also introduces a circle \( D_n \) with the same area as \( P_n \), and then defines \( P_{n+1} \) such that it has the same perimeter as \( D_n \).

The questions to solve are:

1. **For \( n \geq 4 \), express \( \frac{a_{n-1}}{a_n} \) in terms of \( n \).**
   
2. **Find the limit:** 
   \[
   \lim_{n \to \infty} n^2 \left( \frac{a_{n-1}}{a_n} - \frac{n}{\pi} \sin \frac{\pi}{n} \right).
   \]

Would you like me to begin with the first part by deriving the expression for \( \frac{a_{n-1}}{a_n} \)?

正n角形 P_n を次のようにして定義する。 (i) P_3 は面積が1の正三角形である。 (ii) P_n と同じ面積をもつ円を D_n とする。P_{n+1} は D_n と周の長さが等しい正n+1角形である。 n = 3, 4, 5, ... について P_n の面積を a_n としたとき次の各問いに答えよ。 (1) n ≥ 4 について a_{n-1}/a_n をnを用いて表せ。 (2) 極限 lim n→∞ n^2 (a_{n-1}/a_n - n/π sin(π/n)) を求めよ。

Solve a challenging mathematical problem involving the area of regular polygons P_n, the relationship between areas a_n, and explore the limit of an expression involving trigonometric functions as n approaches infinity.

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