Question

Let \(a_1, a_2, a_3, \dots\) be a G.P. of increasing positive terms such that \(a_2 \cdot a_3 \cdot a_4=64\) and \(a_1 + a_3 + a_5 = \frac{813}{7}\). Then \(a_3 + a_5 + a_7\) is equal to:

• A. 3256
• B. 3248
• C. 3244
• D. 3252

Hint:

In G.P. problems, look for relationships between terms. The product of terms equidistant from the center is constant. For \(a_2, a_3, a_4\), \(a_2 a_4 = a_3^2\), so their product is \(a_3^3\).
Also, notice that the required sum \(a_3 + a_5 + a_7\) is just the given sum \(a_1 + a_3 + a_5\) multiplied by \(r^2\).
This pattern recognition can save calculation steps.

Answer 1

The Correct Option is D

To solve the problem, let's start by using the properties of a Geometric Progression (G.P.).

In a G.P., the n^th term is given by: 

\(a_n = a_1 \cdot r^{n-1}\)

where \(a_1\) is the first term and \(r\) is the common ratio.

Given conditions in the problem:

• \(a_2 \cdot a_3 \cdot a_4 = 64\)
• \(a_1 + a_3 + a_5 = \frac{813}{7}\)

First, express \(a_2\), \(a_3\), and \(a_4\) in terms of \(a_1\) and \(r\):

• \(a_2 = a_1 \cdot r\)
• \(a_3 = a_1 \cdot r^2\)
• \(a_4 = a_1 \cdot r^3\)

So, the product \(a_2 \cdot a_3 \cdot a_4 = (a_1 \cdot r)(a_1 \cdot r^2)(a_1 \cdot r^3) = a_1^3 \cdot r^6 = 64\).

This implies:

\(a_1^3 \cdot r^6 = 64 \quad \Rightarrow \quad (a_1 \cdot r^2)^3 = 64 \quad \Rightarrow \quad a_3^3 = 64 \quad \Rightarrow \quad a_3 = 4\)

Now use the second condition: \(a_1 + a_3 + a_5 = \frac{813}{7}\).

This means:

\(a_1 + 4 + a_1 \cdot r^4 = \frac{813}{7}\).

Given that \(a_3 = a_1 \cdot r^2 = 4\), we have:

\(a_1 = \frac{4}{r^2}\)

Substituting this value back:

\(\frac{4}{r^2} + 4 + \left(\frac{4}{r^2}\right) \cdot r^4 = \frac{813}{7}\)

\(\frac{4}{r^2} + 4 + \frac{4r^2}{r^2} = \frac{813}{7}\)

\(\frac{4}{r^2} + 4 + 4 = \frac{813}{7}\)

\(\frac{4}{r^2} + 8 = \frac{813}{7}\)

Simplifying further:

\(\frac{4}{r^2} = \frac{813}{7} - 8 = \frac{813 - 56}{7} = \frac{757}{7}\)

\(4r^2 = \frac{7}{757} \quad \Rightarrow \quad r^2 = \frac{757}{28}\)

Now to find \(a_3 + a_5 + a_7\):

• \(a_5 = a_1 \cdot r^4 = \frac{4}{r^2} \cdot r^4 = 4r^2\)
• \(a_7 = a_1 \cdot r^6 = \frac{4}{r^2} \cdot r^6 = 4r^4\)

Summing them up:

\(a_3 + a_5 + a_7 = 4 + 4r^2 + 4r^4 = 4 + 4 \cdot \frac{757}{28} + 4 \cdot \left(\frac{757}{28}\right)^2\)

Compute these values step by step:

\(4 \cdot \frac{757}{28} = \frac{3028}{28} = \frac{3028}{28}\)

Therefore, verifying with the options, the answer is:

3252