Question

find the common ratio of the geometric sequence: 2, 6, 18, 54, ... the common ratio is \boxed{}.

Explanation:

Step1: Recall the formula for common ratio in a geometric sequence.

The common ratio \( r \) of a geometric sequence is given by \( r=\frac{a_{n + 1}}{a_{n}} \), where \( a_{n+1} \) is the next term and \( a_{n} \) is the current term.

Step2: Calculate the ratio between the second and first term.

Take the second term \( a_{2}=6 \) and the first term \( a_{1} = 2\). Then \( r=\frac{a_{2}}{a_{1}}=\frac{6}{2}=3 \). We can verify with other terms: \(\frac{18}{6} = 3\) and \(\frac{54}{18}=3\), so the common ratio is consistent.

Answer:

3