Discussion
The 3rd and 6th terms of a G.P. are, respectively 108 and -32 . Then the first term is
Answer: C
The first term is 243.
For a G.P. with first term a and ratio r, the n-th term is \(ar^{n-1}\).
\[ \begin{aligned} ar^{2} &= 108,\quad ar^{5} = -32 \\ r^{3} &= \frac{-32}{108} = -\frac{8}{27} \Rightarrow r = -\frac{2}{3} \\ a &= \frac{108}{r^{2}} = 108 \times \frac{9}{4} = 243 \end{aligned} \]
Check: \(243 \times \frac{4}{9} = 108\). The sign of r is negative, but a depends on \(r^{2}\), so a is positive; -243 is the sign trap.
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