Question

what is the common ratio for the geometric sequence?
32, 8, 2, \\(\frac{1}{2}\\), ...
enter your answer by filling in the boxes.
\\(\frac{1}{4}\\)

Explanation:

Step1: Recall the formula for common ratio

The common ratio \( r \) of a geometric sequence is found by dividing any term by its previous term, i.e., \( r=\frac{a_{n}}{a_{n - 1}} \) for \( n\geq2 \).

Step2: Calculate the ratio using the first two terms

Take the second term \( a_{2} = 8 \) and the first term \( a_{1}=32 \). Then \( r=\frac{8}{32}=\frac{1}{4} \). We can verify with other terms: \( \frac{2}{8}=\frac{1}{4} \) and \( \frac{\frac{1}{2}}{2}=\frac{1}{4} \), so the common ratio is consistent.

Answer:

\(\frac{1}{4}\)