Discussion
Let \(a_{1}, a_{2}, a_{3}, ...\) be in G.P. If \(a_{1} . a_{2} . a_{3} = 64\) and \(a_{1} . a_{2} . a_{3} . a_{4} . a_{5} = 32\) , then common ratio is
Answer: D
The common ratio is \(\frac{1}{2}\).
In a G.P. the product of terms placed symmetrically about a middle term is a power of that middle term:
\[ \begin{aligned} a_{1}a_{2}a_{3} &= a_{2}^{3} = 64 \Rightarrow a_{2} = 4 \\ a_{1}a_{2}a_{3}a_{4}a_{5} &= a_{3}^{5} = 32 \Rightarrow a_{3} = 2 \\ r &= \frac{a_{3}}{a_{2}} = \frac{2}{4} = \frac{1}{2} \end{aligned} \]
Check: the G.P. is 8, 4, 2, 1, 1/2, with products 64 and 32. The values 1/4 and 1/8 would make the five-term product far smaller than 32.
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