Reverse Functional Equations

Algebra Level 4

\[ \large f(n) = \begin{cases} 1 & n = 1 \\ \frac{(n-1)!^2}{(n-2)!} & n \geq 2 \end{cases}\]

Define the function \(f \colon \mathbb{N} \to \mathbb{R}\) as shown above.

Which one of the following statements is correct? \(1: f(n+1) = \big( f(n) + f(n-1) + \dots + f(1) \big)\) \(2: f(n+1) = n \big( f(n) + f(n-1) + \dots + f(1) \big)\) \(3: f(n+1) = n^2 \big( f(n) + f(n-1) + \dots + f(1) \big)\) \(4: f(n+1) = \frac{n+1}{n} \big( f(n) + f(n-1) + \dots + f(1) \big)\)

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