Question

If a,b,c are in A.P a,b,d are in G.P,Show that a,a-b,d-c are in G.P.​

Answer 1

Answer:

a, b and c are in AP therefore:

b = a + c / 2

2b = a + c   - (i)

a, b and d are in GP therefore:

b = √ad

b² = ad     -   (ii)

now multiply eq (i) with a

=>  2ab = a² + ac

=>  -ac = a² - 2ab

=>  b² - ac = a² - 2ab + b²

=>  b² - 2ac = (a+b)²

=>  ad - ac = (a+b)²    (from eq (ii))

=>  a (d-c) = (a+b)²

=>  a, a-b, and d-c are in GP

     Hence, Proved

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