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If a ,b ,a n dc be in G.P. and a+x ,b+x ...

If `a ,b ,a n dc` be in G.P. and `a+x ,b+x ,and c+x` in H.P. then find the value of x(a ,b and c are distinct numbers) .

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a+x,b+x, and c+x are in H.P.
`rArrb+x=(2(a+x)(c+x))/((a+x)(c+x))`
or (b+x)(a+c+2x)=2(a+x)(c+x)
or (a+c+2b)x+`2x^(2)+ab+bc=2ac+2x(a+c)+2x^(2)`
or x(c+a-2b)=bc+ab-2ac
or `x(c+a-2b)=bc+ab-2b^(2)` (`because` a,b,c are in G.P)
or x(c+a-2b)=b(c+a-2b)
or x=b (as a,b,c are in G.P. and distinct hence a,b,c cannot be in A.P)
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