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

Explanation

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)