In Exercises 17–24, 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 the first number given. Here, the first term \( a_1 \) is 18.
2. Find the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{6}{18} \).
3. Write the general formula for the nth 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. To find the seventh term \( a_7 \), substitute \( n = 7 \) into the formula and simplify the expression (without calculating the final numeric value).

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

Once the general term formula is established, substitute the desired term number (n) into the formula to find that specific term. For example, to find the seventh term a_7, plug n = 7 into the formula and simplify to get the value.