Question

Let A₁, A₂, A₃, .... be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = `1/1296` and A₂ + A₄ = `7/36`, then the value of A₆ + A₈ + A₁₀ is equal to ______. 

Options

• 33

• 37

• 43

• 47

Answer 1

Let A₁, A₂, A₃, .... be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = `1/1296` and A₂ + A₄ = `7/36`, then the value of A₆ + A₈ + A₁₀ is equal to 43. 

Explanation:

Given relation is A₁ . A₃ . A₅ . A₇ = `1/1296`

Here, A₄ = A₁r³, ⇒ A₁ = `A_4/r^3`.

Then, A₃ = A₁r²

A₃ =  `A_4/r`, A₅ = A₁r₄, A₅ = A₄r, A₇ = A₁r⁶, A₇ = A₄r³

(A₄)⁴ = `1/1296`; A₄ = `1/6`  ......(i)

Take A₂ + A₄ = `7/36`

Put the value of A₂, A₄ = `1/36`  ......(ii)

Divide (i) by (ii),

r² = 6

Now, A₆ = A₄r² = `1/6 xx 6` = 1, A₆ = 1

Similarly, A₈ = 6, A₁₀ = 36

Required is sum A₆ + A₈ + A₁₀ = 43