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

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Solution

Given : a, b, c are in A.P.

⇒ 2b = a + c .......(1)

⇒ $(2 b)^{2} = (a + c)^{2}$

⇒ $4 b^{2} = a^{2} + c^{2} + 2 a c$ ....(2)

Also,

a, b, c are in G.P.

⇒ $b^{2} = a c$ ......(3)

Fron (2) and (3) we get

$4 a c = a^{2} + c^{2} + 2 a c$

⇒ $a^{2} + c^{2} + 2 a c - 4 a c$ = 0

⇒ $a^{2} + c^{2} - 2 a c = 0$

⇒ $(a - c)^{2}$ = 0

⇒ a - c = 0

⇒ a = c ....(4)

From (1) and (4) we get

2b = a + a = c + c

⇒ 2b = 2a = 2c

⇒ a = b = c

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