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Find f(1996)

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,753
If \(\displaystyle f(1)=1996\), \(\displaystyle f(1)+f(2)+\cdots+f(n)=n^2f(n)\), find \(\displaystyle f(1996)\).
 

Albert

Well-known member
Jan 25, 2013
1,225
If \(\displaystyle f(1)=1996\), \(\displaystyle f(1)+f(2)+\cdots+f(n)=n^2f(n)\), find \(\displaystyle f(1996)\).
let f(1)=k=1996
\(\displaystyle f(1)+f(2)+\cdots+f(n-1)=n^2f(n)-f(n)=(n-1)^2f(n-1)\)
$\therefore f(n)=\dfrac{n-1}{n+1}\times f(n-1)$
$ f(1996)=\dfrac {(k-1)(k-2)(k-3)-------(3)(2)(1)}{{(k+1)}(k)(k-1)----(5)(4)(3)}\times f(1)=\dfrac {2}{k+1}$
$=\dfrac {2}{1997}$