Answer :

(Given data in the question) (1)


Cross multiplying (1) and expanding,


(a + bx)(b – cx) = (b + cx)(a-bx)


ab – acx + b2x – bcx2 = ba –b2x + acx – bcx2


2b2x = 2acx


b2 = ac (i)


If three terms are in GP, then the middle term is the Geometric Mean of first term and last term.


b2 = ac


So, from (i) b, is the geometric mean of a and b.


So, a, b, c are in GP.


(Given data in the question) (2)


Cross multiplying (2) and expanding,


(b + cx)(c – dx) = (c + dx)(b – cx)


bc – bdx + c2x – cdx2 = cb – c2x + bdx – dcx2


2c2x = 2bdx


c2 = bd (ii)


So, from (ii), c is the geometric mean of b and d.


So, b, c, d is in GP.


a, b, c, d are in GP.


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