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Monday, February 21, 2022

2022/025) if a-\frac{1}{a} = b, b-\frac{1}{b} = c, c-\frac{1}{c} = a evaluate \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}

 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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