We are given a-\frac{1}{a} = b\cdots(1)
b-\frac{1}{b} = c \cdots(2)
c-\frac{1}{c} = a\cdots(3)
From (1)\frac{1}{a} = a-b
dividing both sides by b
\frac{1}{ab} = \frac{a}{b} - 1 = a(b-c) -1 using |(2)
similarly
\frac{1}{bc} = \frac{b}{c} - 1 = b(c-a) -1
and \frac{1}{ca} = \frac{c}{a} - 1 = -c(a-b) = 1
adding above 3 we have \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}= - 3
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