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f(1998) (Posted on 2008-10-31) Difficulty: 2 of 5
Let f : N --> R a function such that:

f(1) = 999 and
f(1) + f(2) + ... + f(n) = n2*f(n)

for all positive integer n.

Evaluate f(1998).

See The Solution Submitted by pcbouhid    
Rating: 3.0000 (1 votes)

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Solution computer solution (spoiler) | Comment 1 of 4

   10      F=999:N=1:Tot=F
   20 
   25   open "f1998.txt" for output as #2
   30    for N=2 to 2000
   40      F=Tot//(N*N-1)
   50      Tot=Tot+F
   60      print #2,N,F
   70    next
   80   close

produces (shown with ellipsis for space considerations--also UBASIC's doubled slashes reduced to single):


 2   333
 3   333/2
 4   999/10
 5   333/5
 6   333/7
 7   999/28
 8   111/4
 9   111/5
 10   999/55
 11   333/22
 12   333/26
 13   999/91
 14   333/35
 15   333/40
 16   999/136
 17   111/17
 18   111/19
 19   999/190
 20   333/70
 21   333/77
 22   999/253
 23   333/92
 24   333/100
 25   999/325
 26   37/13
 27   37/14
 28   999/406
 29   333/145
 30   333/155
 31   999/496
 32   333/176
 33   333/187
 34   999/595
 35   111/70
 36   3/2
 37   27/19
 38   333/247
 39   333/260
 40   999/820
 41   333/287
 42   333/301
 43   999/946
 44   111/110
 45   111/115
 46   999/1081
 47   333/376
 48   333/392
 49   999/1225
 50   333/425
 51   333/442
 52   999/1378
 53   37/53
 54   37/55
 55   999/1540
 56   333/532
 57   333/551
 58   999/1711
 59   333/590
 60   333/610
 61   999/1891
 62   111/217
 63   111/224
...
 1976   333/651092 
 1977   333/651751
 1978   999/1957231
 1979   111/217690
 1980   111/217910
 1981   999/1963171
 1982   333/655051
 1983   333/655712
 1984   999/1969120
 1985   333/657035
 1986   333/657697
 1987   999/1975078
 1988   111/219674
 1989   111/219895
 1990   999/1981045
 1991   333/661012
 1992   333/661676
 1993   999/1987021
 1994   333/663005
 1995   333/663670
 1996   999/1993006
 1997   1/1997
 1998   1/1999
 1999   999/1999000
 2000   333/667000

Inspection shows f(n) = 999 * 2 / (n*(n+1)), but in any event, f(1998) = 1/1999.


  Posted by Charlie on 2008-10-31 14:23:37
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