If a, b and c are in A.P. and also in G.P., show that : a = b = c .
Given : a, b, c are in A.P.
⇒ 2b = a + c .......(1)
⇒ (2b)2=(a+c)2
⇒ 4b2=a2+c2+2ac ....(2)
Also,
a, b, c are in G.P.
⇒ b2=ac ......(3)
Fron (2) and (3) we get
4ac=a2+c2+2ac
⇒ a2+c2+2ac−4ac = 0
⇒ a2+c2−2ac=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