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If a, b, c are in AP or GP or HP, then ...

If `a, b, c` are in AP or GP or HP, then `(a-b)/(b-c)` is equal to

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Given that ,a,b, and c are in GP .
then `(b)/(a)=( c) /(b)=r` [constant]
`implies b=arimplies c=br`
`therefore (a-b)/(b-c) =-(a-ar)/(ar-br)=(a(1-r))/(r(a-b))=(a(1-r))/(r(a-ar))`
`=(a(1-r))/(ar(1-r))=(1)/(r ) `
` therefore (a-b)/(b-c)= (1)/(r ) =(a) /(b) or (b)/(c)`
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