Q15 of 176 Page 20

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

a, b, c are in AP


So, 2b = a + c …(1)


b, c, d are in GP


So, b2 = ad …(2)


Multiply first equation with a and subtract it from 2nd.


b2 – 2ab = ad – ac – a2


a2 + b2 – 2ab = a(d – c)


⇒ (a – b)2 = a(d – c)


As a, (a – b), (d – c) satisfy the geometric mean relationship


Hence a, (a – b),(d – c) are in G.P.


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