If a, b, c are in A.P. b, c, d are in G.P. and are in A.P., prove that a, c, e are in G.P.

Given:


a,b,c are in AP


2b = a + c …… (i)


b,c,d are in GP;


c2 = bd …… (ii)


1/c, 1/d, 1/e are in AP;



…(iii)


From the above substituting for b & d in (ii) above,



c(c + e) = (a + c) e


c2 + ce = ae + ce


c2 = ae


Thus a, c, e are in GP


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