Discussion
The sum of the second and third terms of a G.P. is 8 and the fourth term is 4. The common ratio \(r \neq 1\) is
Answer: A
The common ratio is \(-\frac{1}{2}\).
Write the terms as \(ar, ar^{2}, ar^{3}\):
\[ \begin{aligned} ar(1 + r) &= 8, \quad ar^{3} = 4 \\ \frac{1 + r}{r^{2}} &= 2 \\ 2r^{2} - r - 1 &= 0 \\ (2r + 1)(r - 1) &= 0 \end{aligned} \]
So r = 1 or \(r = -\frac{1}{2}\), and since \(r \neq 1\), \(r = -\frac{1}{2}\). Check: a = -32 gives terms -32, 16, -8, 4; 16 + (-8) = 8 and the fourth term is 4.
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