Question

If a, b, c is in A.P. then show that 3^a, 3^b, 3^c  is in G.P.

Answer 1

If a, b, c are in A.P

t₂ – t₁ = t₃ – t₂

b – a = c – b

2b = c + a

To prove that 3^a, 3^b, 3^c is in G.P

⇒ 3^2b = 3^c+a + a  ...[Raising the power both sides]

⇒ 3^b. 3^b = 3^c. 3^a 

⇒ `(3^"b")/(3^"a") = (3^"c")/(3^"b")`

⇒ `("t"_2)/("t"_1) = ("t"_3)/("t"_1)`

⇒ Common ratio is same for 3^a, 3^b, 3^c 

⇒ 3^a, 3^b, 3^c forms a G.P

∴ Hence it is proved.