Question

write an equation for the nth term of the geometric sequence 2, 6, 18, 54, ...

Explanation:

Step1: Recall geometric sequence formula

The formula for the \( n \)-th term of a geometric sequence is \( a_n = a_1 \cdot r^{n - 1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step2: Identify \( a_1 \) and \( r \)

For the sequence \( 2, 6, 18, 54, \dots \), the first term \( a_1 = 2 \). The common ratio \( r \) is \( \frac{6}{2} = 3 \) (or \( \frac{18}{6} = 3 \), etc.).

Step3: Substitute into the formula

Substitute \( a_1 = 2 \) and \( r = 3 \) into \( a_n = a_1 \cdot r^{n - 1} \). We get \( a_n = 2 \cdot 3^{n - 1} \).

Answer:

\( a_n = 2 \cdot 3^{n - 1} \)