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Saturday, May 14, 2022

2022/037) If a, b, c are in H.P., how do you prove that \frac{b+c-a}{a}, \frac{c+a-b}{b}, \frac{a+b-c}{c} are in A.P.​?

 a,b,c are in HP so \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in AP

or \frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}

multiply both LHS and RHS by a + b + c we get

\frac{a+b+c}{b} - \frac{a+b+c}{a} = \frac{a+b+c}{c} - \frac{a+b+c}{b}

Or 

(\frac{a+b+c}{b} - 2) - (\frac{a+b+c}{a}-2)  = (\frac{a+b+c}{c} -2)  - (\frac{a+b+c}{b} -2)

or (\frac{a+c-b }{b}) - (\frac{b+c-a }{a})  = (\frac{a+b-c }{c})  - (\frac{c + a - b}{b} )

Hence  \frac{b+c-a}{a}, \frac{c+a-b}{b}, \frac{a+b-c}{c} are in A.P

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