In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=2n/(n+4)

Verified step by step guidance

1. Identify the general term of the sequence: \(a_n = \frac{2n}{n+4}\).
2. To find the first term \(a_1\), substitute \(n = 1\) into the general term: \(a_1 = \frac{2(1)}{1+4}\).
3. To find the second term \(a_2\), substitute \(n = 2\) into the general term: \(a_2 = \frac{2(2)}{2+4}\).
4. To find the third term \(a_3\), substitute \(n = 3\) into the general term: \(a_3 = \frac{2(3)}{3+4}\).
5. To find the fourth term \(a_4\), substitute \(n = 4\) into the general term: \(a_4 = \frac{2(4)}{4+4}\).

Key Concepts

Sequences

A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and the position of a term is typically denoted by an index, such as 'n'. Understanding how to identify and generate terms from a given formula is essential for working with sequences.

General Term of a Sequence

The general term of a sequence, often denoted as an, is a formula that defines the nth term of the sequence in terms of n. In this case, the general term is given by an = 2n/(n+4). This formula allows us to calculate any term in the sequence by substituting different values of n.

Substitution

Substitution is the process of replacing a variable in an expression with a specific value. To find the first four terms of the sequence defined by the general term, we substitute n = 1, 2, 3, and 4 into the formula an = 2n/(n+4). This step is crucial for generating the actual terms of the sequence.