Question

What is a possible value for the missing term of the geometric sequence?

1250, ___, 50, ...

Answer 1

There's a formal way to do this, and then there's a hacky way. I'd introduce both, in any case.

Answer 2

Formal way first:\[\begin{equation}a = 1250 \\ ar^2 = 50 \end{equation} \]Dividing,\[\dfrac{ar^2}{a} =\dfrac{50}{1250} \Rightarrow r^2 = 1/25 \Rightarrow r = \pm 1/5 \]The second term is the first term times the common ratio, so \(t_2 = 1250 \times 1/5\) OR \(t = 1250 \times -1/5\).

Answer 3

Hacky way: if three terms are in geometric sequence, then the middle one can be the geometric mean of the first and third. What is the geometric mean of 1250 and 50?