Question

If a,b,c are in AP as well as in GP then show that a=b=c.​

Answer 1

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

Answer 2

Answer:

Step-by-step explanation:

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

HOPE IT HELPS..!!