Question

a student wrote the first three values in a geometric sequence as shown below.
f, g, h, ...
which of the following shows the correct relationship between these terms?
f - g = g - h
fg = gh
h = \frac{g}{f}
h = \frac{g^2}{f}

Explanation:

Step1: Recall Geometric Sequence Definition

In a geometric sequence, the ratio between consecutive terms is constant. So, $\frac{g}{f} = \frac{h}{g}$.

Step2: Derive Relationship

From $\frac{g}{f} = \frac{h}{g}$, cross - multiply to get $g\times g=h\times f$, which simplifies to $h = \frac{g^{2}}{f}$.

Step3: Analyze Other Options

• For the option $f - g=g - h$, this is the property of an arithmetic sequence (common difference), not geometric.
• For the option $fg = gh$, dividing both sides by $g$ (assuming $g

eq0$) gives $f = h$, which is not a general property of a geometric sequence.

• For the option $h=\frac{g}{f}$, from the geometric sequence ratio $\frac{g}{f}=\frac{h}{g}$, we can see this is incorrect as it would imply $h=\frac{g^{2}}{f}$ not $\frac{g}{f}$.

Answer:

D. $h=\frac{g^{2}}{f}$ (assuming the options are labeled A, B, C, D with the last option being $h = \frac{g^{2}}{f}$)