Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

9. Sequences, Series, & Induction

Sequences

College Algebra

9. Sequences, Series, & Induction

Sequences

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2:29 minutes

Problem 9

Textbook Question

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

Identify the general term of the sequence: $a_n = \frac{2n}{n+4}$.

To find the first term $a_1$, substitute $n = 1$ into the general term: $a_1 = \frac{2(1)}{1+4}$.

To find the second term $a_2$, substitute $n = 2$ into the general term: $a_2 = \frac{2(2)}{2+4}$.

To find the third term $a_3$, substitute $n = 3$ into the general term: $a_3 = \frac{2(3)}{3+4}$.

To find the fourth term $a_4$, substitute $n = 4$ into the general term: $a_4 = \frac{2(4)}{4+4}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.