Discussion
If \(a, x, y, b\) are in G.P. , then \((x + y)^{2} =\)
Answer: A
\((x + y)^{2}\) equals \((a + x)(y + b)\).
Let the common ratio be r, so \(x = ar\), \(y = ar^{2}\), \(b = ar^{3}\).
\[ \begin{aligned} (x + y)^{2} &= a^{2}r^{2}(1 + r)^{2} \\ (a + x)(y + b) &= a(1 + r) \cdot ar^{2}(1 + r) \\ &= a^{2}r^{2}(1 + r)^{2} \end{aligned} \]
The two agree. A quick check with 1, 2, 4, 8: \((2 + 4)^{2} = 36\) and \((1 + 2)(4 + 8) = 36\), while \((8 + 2)(4 + 1) = 50\) rules out the second option and the doubled terms rule out the rest.
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