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

Text Solution

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

`(a)/(b)`

`(b)/(a)`

`(a)/(c)`

`(c)/(b)`

If a,b,c are in AP and b,c,d be in HP, then

`ab=cd`

`ad=bc`

`ac=bd`

`abcd=1`

If a,b,c are in GP, a-b,c-a,b-c are in HP, then a+4b+c is equal to

`0`

`-1`

None of these