Question

Let $a, \frac{3}{4}, ar^{2}, ar^{3}, \dots$ be in G.P. where $r>0$. If the product of the first four terms is $\frac{3^{6}}{4^{5}}$, then $a$ is equal to ________.

• A. $\frac{3}{2}$
• B. $\frac{2}{3}$
• C. $\frac{1}{3}$
• D. $\frac{1}{2}$
• E. 1

Hint:

$n$-th term of G.P. is $ar^{n-1}$.

Answer 1

The Correct Option is D

Step 1: Concept
In a G.P., terms follow the pattern $a, ar, ar^2, ar^3$.

Step 2: Meaning
The second term is $ar = \frac{3}{4} \implies r = \frac{3}{4a}$.

Step 3: Analysis
Product: $a \cdot (ar) \cdot (ar^2) \cdot (ar^3) = a^4 r^6 = \frac{3^6}{4^5}$. Substitute $r$: $a^4 (\frac{3}{4a})^6 = \frac{3^6}{4^5} \implies \frac{3^6}{4^6 a^2} = \frac{3^6}{4^5}$.

Step 4: Conclusion
$4^5 = 4^6 a^2 \implies a^2 = \frac{1}{4} \implies a = \frac{1}{2}$ (since $r>0$). Final Answer: (D)