In Exercises 31–34, write the first five terms of each geometric sequence. a1 = 3, r = 2

Verified step by step guidance

1. Identify the first term of the geometric sequence, which is given as \( a_1 = 3 \).
2. Recognize that the common ratio \( r \) is given as 2.
3. Use the formula for the \( n \)-th term of a geometric sequence: \( a_n = a_1 \cdot r^{n-1} \).
4. Calculate the second term: \( a_2 = 3 \cdot 2^{2-1} = 3 \cdot 2 \).
5. Continue this process to find the next three terms: \( a_3 = 3 \cdot 2^{3-1} \), \( a_4 = 3 \cdot 2^{4-1} \), and \( a_5 = 3 \cdot 2^{5-1} \).

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. This type of sequence can be expressed in the form a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.

First Term (a1)

The first term of a geometric sequence, denoted as a1, is the initial value from which the sequence begins. In the given problem, a1 = 3 indicates that the first term of the sequence is 3, which serves as the foundation for calculating subsequent terms using the common ratio.

Common Ratio (r)

The common ratio, denoted as r, is the factor by which each term in a geometric sequence is multiplied to obtain the next term. In this case, r = 2 means that each term will be double the previous term, allowing for the generation of the sequence's subsequent values based on the first term.