Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a₇, the seventh term of the sequence. 18, 6, 2, 2/3, ...

Verified step by step guidance

1. Identify the first term of the geometric sequence, which is \(a_1 = 18\).
2. Find the common ratio \(r\) by dividing the second term by the first term: \(r = \frac{6}{18}\).
3. Write the formula for the general term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\).
4. Substitute the values of \(a_1\) and \(r\) into the formula to get the explicit formula for \(a_n\).
5. Use the formula to find the seventh term by substituting \(n = 7\) into \(a_n = a_1 \times r^{n-1}\).

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the sequence 18, 6, 2, 2/3, ..., each term is obtained by multiplying the previous term by 1/3.

General Term Formula of a Geometric Sequence

The general term (nth term) of a geometric sequence is given by a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number. This formula allows you to find any term in the sequence without listing all previous terms.

Evaluating the nth Term

To find a specific term like the seventh term (a_7), substitute n = 7 into the general term formula. Calculate the power of the common ratio and multiply by the first term to get the value of the term directly.