Q1 of 176 Page 21

Insert 6 geometric means between 27 and .

Let the six terms be a₁, a₂, a₃, a₄, a₅, a₆.

And,

A = 27, B

Now, these 6 terms are between A and B.

∴ A, a₁, a₂, a₃, a₄, a₅, a₆, B.

Now all of them are in GP

So we now have 8 terms in GP with the first term being 27 and eighth being 1/81.

We know that, T_n = arⁿ^–1

Here T_n = , a = 27 and

⇒

∴ a₁ = Ar =

a₂ = Ar² =

a₃ = Ar³ =

a₄ = Ar⁴ =

a₅ = Ar⁵ =

a₆ = Ar⁶ =

∴ The six GM between 27 and 1/81 are .

If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x < – 1 or x > 3.

If p^th, q^th and r^th terms of an A.P. and G.P. are both a, b and c respectively, show that a^{b – c}b^c ^{– a} c^{a – b} = 1.

Insert 5 geometric means between and .