实现功能：计算斐波那契数列的第n项

以下是用 Python 编写的递归方法实现斐波那契数列：

def fibonacci(n): if n <= 0: return "输入必须为正整数" elif n == 1: return 0 elif n == 2: return 1 else: return fibonacci(n - 1) + fibonacci(n - 2) # 示例调用 n = 10 print(f"斐波那契数列的第{n}项是: {fibonacci(n)}")

迭代方法优化

递归方法在 n 较大时效率较低，可以使用迭代方法优化：

def fibonacci_iterative(n): if n <= 0: return "输入必须为正整数" a, b = 0, 1 for _ in range(2, n): a, b = b, a + b return b if n > 1 else a # 示例调用 n = 10 print(f"斐波那契数列的第{n}项是: {fibonacci_iterative(n)}")

动态规划方法

使用动态规划存储中间结果，避免重复计算：

def fibonacci_dp(n, memo={}): if n <= 0: return "输入必须为正整数" if n in memo: return memo[n] if n == 1: return 0 elif n == 2: return 1 memo[n] = fibonacci_dp(n - 1, memo) + fibonacci_dp(n - 2, memo) return memo[n] # 示例调用 n = 10 print(f"斐波那契数列的第{n}项是: {fibonacci_dp(n)}")

公式法（Binet公式）

对于大数计算，可以使用数学公式近似计算：

import math def fibonacci_formula(n): if n <= 0: return "输入必须为正整数" sqrt5 = math.sqrt(5) phi = (1 + sqrt5) / 2 return round((phi ** n - (-phi) ** (-n)) / sqrt5) # 示例调用 n = 10 print(f"斐波那契数列的第{n}项是: {fibonacci_formula(n)}")

以上方法可根据实际需求选择，递归适合小规模计算，迭代和动态规划适合高效计算，公式法适用于大数近似计算。

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