Question

Insert 5 geometric means between 32/9 and 81/2.

Answer 1

Let the five terms be a₁, a₂, a₃, a₄, a₅.

A = 32/9, B = 81/2

Now, these 5 terms are between A and B.

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

So we now have 7 terms in GP with the first term being 32/9 and seventh being 81/2.

We know that, T_n = ar^n–1

Here, T_n = 81/2, a = 32/9 and

81/2 = 32/9r^7-1

(81 × 9)/(2 × 32) = r⁶

r = 3/2

a₁ = Ar = (32/9) × 3/2 = 16/3 

a₂ = Ar² = (32/9) × 9/4 = 8   

a₃ = Ar³ = (32/9) × 27/8 = 12   

a₄ = Ar⁴ = (32/9) × 81/16 = 18   

a₅ = Ar⁵ = (32/9) × 243/32 = 27 

∴ The five GM between 32/9 and 81/2 are 16/3, 8, 12, 18, 27