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let f : N --> R be function such that f(1) + 2f(2) + 3f(3) + ............... + nf(n) = n(n+1)f(n), for n>= 2 and f(1) = 1 then find the valueof f(n)and f(5) is ?

let f : N --> R be function such that f(1) + 2f(2) + 3f(3) + ............... + nf(n) = n(n+1)f(n), for n>= 2 and f(1) = 1 then find the valueof f(n)and f(5) is ?

Grade:12th pass

2 Answers

Meet
137 Points
6 years ago
First put the value of n=2 and then put f(1)=1 then u will get f(2)=1/4 and similarly If you put n=3 then u will get f(3)=1/6 and similarly f(4)=1/8, f(5)=1/10, so therefore by this we can say that f(n) =1/n for n>_2
mycroft holmes
272 Points
6 years ago
We have f(1)+2f(2)+...+nf(n) = n(n+1)f(n)
 
For k=n+1 we get
f(1)+2f(2)+...+n f(n)+(n+1) f(n+1) = (n+1)(n+2)f(n+1)
 
Subtracting the two we get (n+1)(n+2) f(n+1)-n(n+1)f(n)=(n+1) f(n+1)
 
or (n+1)(n+2) f(n+1) – (n+1) f(n+1) = n(n+1)f(n)
 
or (n+1)2 f(n) = (n+1) f(n)
 
or (n+1)f(n+1) = n f(n)
 
Hence we get that for all k, kf(k) = C where C is a constant. When k=1, we see that 
C=1. So that f(k) = 1/k for all k

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