Let f(x) is an odd function on R, f(1)=1 and f(x/(x-1))=xf(x) for all x<0. Find the value of
f(1)f(1/100)+f(1/2)f(1/99)+f(1/3)f(1/98)+...+f(1/50)f(1/51).
f(1/(n+1)) = f((-1/n)/(-(n+1)/n)) = f((-1/n)/(-1/n - 1)) just algebra
= -(1/n)f(-1/n) see rule above
= (1/n)f(1/n) because an odd function
Then,
f(1) = 1
f(1/2) = 1/1 * f(1) = 1
f(1/3) = 1/2 * f(1/2) = 1/2
f(1/4) = 1/3* f(1/3) = 1/6
f(1/5) = 1/4 * f(1/4) = 1/24
In general, if n is a positive integer
f(1/n) = 1/(n-1)! provable by induction
I do not see an obvious way to do the requested sum, so this is as far as I go
A start
