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If a ,b ,c are in G.P. and a-b ,c-a ,a n...

If `a ,b ,c` are in G.P. and `a-b ,c-a ,a n db-c` are in H.P., then prove that `a+4b+c` is equal to 0.

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a,b,c in G.P
`rArrb=ar,c=ar^(2)`
`rArr1/(a-b),1/(c-a),1/(b-c)` are in A.P.
`rArr2/(a(r^(2)-1))=1/(a(1-r))+1/(ar(1-r))=(1+r)/(ar(1-r))`
`rArr-2r=(r+1)^(2)`
`rArrr^(2)+4r+1=0` (1)
Also, c+4b+a=`a(r^(2)+4r+1)=0` [Using (1)]
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