13. If b is the harmonic mean between a and c, prove 1 1 1 1 that b -a b-C

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( b=frac{2 a c}{a+c} ) ( frac{L H S}{N O C O} frac{1}{b-a}+frac{1}{b-C}=frac{1}{left(frac{2 a C}{a+C}right)-a}+frac{1}{left(frac{2 a C}{a+C}right)} ) ( =frac{a+c}{2 a c-a(a+c)}+frac{a+c}{2 a c-c(a+c)} ) ( =frac{a+c}{a c-a^{2}}+frac{a+c}{a c-c^{2}} ) ( Rightarrow frac{a+c}{a(c-a)}+frac{a+c}{c(a-c)} Rightarrow frac{a+c}{a(c-a)}+frac{a+c}{c(-c+a)} ) ( Rightarrow quad frac{a+c}{a(c-a)}-frac{a+c}{c(c-a)} ) ( Rightarrowleft(frac{a+c}{c-a}right)left[frac{1}{a}-frac{1}{c}right] Rightarrowleft(frac{a+c}{c-a}right) frac{x(c-a)}{a c} ) ( Rightarrow quad frac{a+c}{a c} quad Rightarrow quad frac{a}{a c}+frac{c}{a c} ) ( Rightarrow quad frac{1}{c} quad+frac{1}{a}=R H S )

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