In Exercises 1–8, write the first five terms of each geometric sequence. a1 = 5, r = 3

Verified step by step guidance

1. Identify the first term of the geometric sequence, which is given as \( a_1 = 5 \).
2. Recognize that the common ratio \( r \) is given as 3.
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: \( a_2 = 5 \cdot 3^{2-1} = 5 \cdot 3 \).
5. Continue calculating the next terms using the formula: \( a_3 = 5 \cdot 3^{3-1} \), \( a_4 = 5 \cdot 3^{4-1} \), and \( a_5 = 5 \cdot 3^{5-1} \).

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 a_1, is the initial value from which the sequence begins. In the given problem, a_1 is specified as 5, meaning that the first term of the sequence is 5. This value is crucial for calculating subsequent terms in the sequence.

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 is given as 3, indicating that each term will be three times the previous term. Understanding the common ratio is essential for generating the terms of the sequence.