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        riddles >> medium >> f(2007)
        
(Message started by: perash on Jan 4^th, 2007, 11:25am)

Title: f(2007)
Post by perash on Jan 4^th, 2007, 11:25am

if f(1)=2007
n is integers
f(1)+f(2)+.......+f(n)=(n^2).f(n)
and n>= 1
find f(2007)

Title: Re: f(2007)
Post by towr on Jan 4^th, 2007, 12:32pm

[hide]f(n) = 2007 / (1/2 n(n+1))

f(2007) = 2/2008 = 1/1004[/hide]

Title: Re: f(2007)
Post by balakrishnan on Jan 6^th, 2007, 10:31am

[hide]f(1)+f(2)+..f(n)=n^2 f(n)
f(1)+..f(n-1)=(n-1)^2 f(n-1)
subtracting the 2,we get
f(n)=n^2 f(n)-(n-1)^2 f(n-1)
which gives
(n^2-1) f(n) =(n-1)^2 f(n-1)
or
(n+1) f(n)=(n-1) f(n-1)
or
f(n)=(n-1)/(n+1) f(n-1)

which means
f(n)=(n-1)/(n+1)*(n-2)/(n)*(n-3)/(n-1)*...(1/3)*f(1)
or
f(n)=[(n-1)!]/[(n+1)!] *2 * f(1)=2*f(1)/[n(n+1)]
which gives
f(2007)=2/2008=1/1004[/hide]