Discussion
If the numbers \(x, 6, y, 54, 162\) are in geometric progression, then \(\frac{y}{x}\) is equal to
Answer: C
\(\frac{y}{x}\) equals 9.
In a geometric progression every term is the previous one times the common ratio r. From the last two terms, \(r = \frac{162}{54} = 3\). Working back: \(y = \frac{54}{3} = 18\), then \(6 = 18/3\) checks, and \(x = \frac{6}{3} = 2\).
\[ \frac{y}{x} = \frac{18}{2} = 9 = r^{2} \]
Since y is two steps after x, \(\frac{y}{x}\) is always \(r^{2}\). 3 is the ratio r itself, and 18 is the value of y.
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