Let f: N rightarrow R be such that f(1)=1 and | vedclass

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(C) Given $f(1)=1$ and $f(1)+2 f(2)+3 f(3)+\ldots+n f(n)=n(n+1) f(n)$ for $n \geq 2$. Let $S_n = \sum_{k=1}^{n} k f(k)$. Then $S_n = n(n+1) f(n)$. For

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$1/500$

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(C) Given $f(1)=1$ and $f(1)+2 f(2)+3 f(3)+\ldots+n f(n)=n(n+1) f(n)$ for $n \geq 2$. Let $S_n = \sum_{k=1}^{n} k f(k)$. Then $S_n = n(n+1) f(n)$. For $n \geq 2$, $S_n = S_{n-1} + n f(n) = n(n+1) f(n)$. Substituting $S_{n-1} = (n-1)n f(n-1)$, we get $(n-1)n f(n-1) + n f(n) = n(n+1) f(n)$. Dividing by $n$ $(n \geq 2)$, we get $(n-1) f(n-1) + f(n) = (n+1) f(n)$. $(n-1) f(n-1) = n f(n) \Rightarrow f(n) = \frac{n-1}{n} f(n-1)$. For $n=2$, $f(2) = \frac{1}{2} f(1) = \frac{1}{2}$. For $n=3$, $f(3) = \frac{2}{3} f(2) = \frac{2}{3} 	imes \frac{1}{2} = \frac{1}{3}$. In general, $f(n) = \frac{1}{n}$. Thus, $f(500) = \frac{1}{500}$.

Let $f: N \rightarrow R$ be such that $f(1)=1$ and $f(1)+2 f(2)+3 f(3)+\ldots+n f(n)=n(n+1) f(n)$ for all $n \in N, n \geq 2,$ where $N$ is the set of natural numbers and $R$ is the set of real numbers. Then, the value of $f(500)$ is

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