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

Since, a, b and c are in A.P.

⇒ 2b = a + c

⇒ b = $\dfrac{a + c}{2}$ …….(i)

Since, a, b and c are also in G.P.

⇒ b² = ac

Substituting value of b from (i) in above equation we get,

$\Rightarrow \Big(\dfrac{a + c}{2}\Big)^2 = ac \\[1em] \Rightarrow \dfrac{a^2 + c^2 + 2ac}{4} = ac \\[1em] \Rightarrow a^2 + c^2 + 2ac = 4ac \\[1em] \Rightarrow a^2 + c^2 - 2ac = 0 \\[1em] \Rightarrow (a - c)^2 = 0\\[1em] \Rightarrow a - c = 0 \\[1em] \Rightarrow a = c …….(ii)$

Substituting value of a from (ii) in (i) we get,

b = $\dfrac{c + c}{2} = \dfrac{2c}{2}$ = c …….(iii)

From (ii) and (iii) we get,

a = b = c.

Hence, proved that a = b = c.