Discussion
The two geometric means between 2 and 8 are \(g_{1}\) and \(g_{2}\) . Then \(g_{1}^{3} + g_{2}^{3}\) is equal to
Answer: A
\(g_{1}^{3} + g_{2}^{3}\) equals 160.
Inserting two geometric means makes 2, \(g_{1}\), \(g_{2}\), 8 a G.P., so \(2r^{3} = 8\) and \(r^{3} = 4\).
\[ \begin{aligned} g_{1}^{3} &= (2r)^{3} = 8 \times 4 = 32 \\ g_{2}^{3} &= (2r^{2})^{3} = 8 \times 16 = 128 \\ g_{1}^{3} + g_{2}^{3} &= 160 \end{aligned} \]
There is no need to find \(r = 4^{1/3}\) itself; cubing the means turns \(r^{3}\) and \(r^{6}\) into 4 and 16. A neat check: \(g_{1}g_{2} = 2 \times 8 = 16\).
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