Question

If a, b and c are in A.P. and also in G.P., show that : a = b = c.

Answer 1

a, b and c are in A.P.

`=>` 2b = a + c

`=> b = (a + c)/2`

a, b and c are also in G.P. 

`=>` b² = ac

`=> ((a + c)/2)^2 = ac`

`=> (a^2 + c^2 + 2ac)/4 = ac`

`=>` a² + c² + 2ac = 4ac

`=>` a² + c² – 2ac = 0

`=>` (a – c)² = 0

`=>` a – c = 0

`=>` a = c

Now, 2b = a + c

`=>` 2b = a + a

`=>` 2b = 2a

`=>` b = a

Thus, we have a = b = c