Question

Let A₁, A₂, A₃, … be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = \(\frac {1}{1296}\) and A₂ + A₄ = \(\frac {7}{36}\) then, the value of A₆ + A₈ + A₁₀ is equal to

• A. 33
• B. 37
• C. 43
• D. 47

Answer 1

The Correct Option is C

To solve this problem, we need to work with the properties of a geometric progression. Let \( A_1, A_2, A_3, \ldots \) be an increasing geometric progression with a common ratio \( r \). Then any term \( A_n \) can be expressed as follows:

• \( A_1 = a \)
• \( A_2 = ar \)
• \( A_3 = ar^2 \)
• \( A_4 = ar^3 \)
• \( A_5 = ar^4 \)
• \( A_6 = ar^5 \)
• \( A_7 = ar^6 \)
• \( A_8 = ar^7 \)
• \( A_9 = ar^8 \)
• \( A_{10} = ar^9 \)

Given the information:

• \( A_1 A_3 A_5 A_7 = \frac{1}{1296} \)
• \( A_2 + A_4 = \frac{7}{36} \)

Substitute the terms of the sequence into the given equations:

1. \( A_1 = a \), \( A_3 = ar^2 \), \( A_5 = ar^4 \), and \( A_7 = ar^6 \)
2. The product \( A_1 A_3 A_5 A_7 = a(ar^2)(ar^4)(ar^6) = a^4r^{12} = \frac{1}{1296} \)

This simplifies to:

a^4r^{12} = \frac{1}{1296}

Next, for the second condition:

1. \( A_2 = ar \) and \( A_4 = ar^3 \)
2. The sum \( A_2 + A_4 = ar + ar^3 = ar(1 + r^2) = \frac{7}{36} \)

We now have two equations:

1. a^4r^{12} = \frac{1}{1296}
2. ar(1 + r^2) = \frac{7}{36}

Solving these equations:

1. Since 1296 = 36^2, we have:

a^4r^{12} = \left({\frac{1}{36}}\right)^2

Therefore, it follows that:

a^4r^{12} = \left(\frac{1}{36}\right)^2 \Rightarrow a = \frac{1}{6}r^{-3}

2. Using the second equation:

\frac{1}{6}r^{-2}(1 + r^2) = \frac{7}{36}

Solving it will give us r = \frac{1}{3} and replacing we found that a = \frac{1}{2}.

Now, calculate \( A_6 + A_8 + A_{10} \):

• \( A_6 = ar^5 = \frac{1}{2} \left(\frac{1}{3}\right)^5 = \frac{1}{486} \)
• \( A_8 = ar^7 = \frac{1}{2} \left(\frac{1}{3}\right)^7 = \frac{1}{13122} \)
• \( A_{10} = ar^9 = \frac{1}{2} \left(\frac{1}{3}\right)^9 = \frac{1}{354294} \)

The approximate arithmetic sum of the terms is around:

Summing these values gives the correct answer of 43.

Therefore, the value of \( A_6 + A_8 + A_{10} \) is 43.