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. 3, 12, 48, 192, ...

Verified step by step guidance

1. <Identify the first term of the sequence, denoted as a_1. In this sequence, a_1 = 3.>
2. <Determine the common ratio (r) of the geometric sequence by dividing the second term by the first term. Here, r = \frac{12}{3} = 4.>
3. <Write the formula for the nth term of a geometric sequence: a_n = a_1 \cdot r^{n-1}.>
4. <Substitute the known values into the formula: a_n = 3 \cdot 4^{n-1}.>
5. <To find the seventh term (a_7), substitute n = 7 into the formula: a_7 = 3 \cdot 4^{7-1}.>

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 3, 12, 48, 192, the common ratio is 4, as each term is obtained by multiplying the previous term by 4.

General Term Formula

The general term (nth term) of a geometric sequence can be expressed using the formula 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 us to calculate any term in the sequence based on its position.

Finding Specific Terms

To find a specific term in a geometric sequence, such as the seventh term (a_7), we substitute n with 7 in the general term formula. By calculating a_7 using the established formula, we can determine the value of the seventh term in the sequence.