Question

Insert 6 geometric means between 27 and 1/81.

Answer 1

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

A = 27, B = 1/81

Now, these 6 terms are between A and B.

So the GP is: A, a₁, a₂, a₃, a₄, a₅, a₆, B.

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

We know that, T_n = ar^n–1

Here, T_n = 1/81, a = 27 and

1/81 = 27r^8-1

1/(81 × 27) = r⁷

r = 1/3

a₁ = Ar = 27 × 1/3 = 9  

a₂ = Ar² = 27 × 1/9 = 3   

a₃ = Ar³ = 27 × 1/27 = 1   

a₄ = Ar⁴ = 27 × 1/81 = 1/3   

a₅ = Ar⁵ = 27 × 1/243 = 1/9   

a₆ = Ar⁶ = 27 × 1/729 = 1/27   

∴ The six GM between 27 and 1/81 are 9, 3, 1, 1/3, 1/9, 1/27