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

Arithmetic Sequences

College Algebra

9. Sequences, Series, & Induction

Arithmetic Sequences

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1:53 minutes

Problem 9

Textbook Question

In Exercises 1–14, write the first six terms of each arithmetic sequence. an = an-1 +6, a₁ = −9

Verified step by step guidance

Identify the first term of the sequence, which is given as \( a_1 = -9 \).

Recognize that the sequence is arithmetic, meaning each term is obtained by adding a constant difference to the previous term. Here, the common difference \( d \) is 6.

Use the recursive formula \( a_n = a_{n-1} + 6 \) to find the second term: \( a_2 = a_1 + 6 \).

Continue using the recursive formula to find the third term: \( a_3 = a_2 + 6 \).

Repeat the process to find the fourth, fifth, and sixth terms: \( a_4 = a_3 + 6 \), \( a_5 = a_4 + 6 \), and \( a_6 = a_5 + 6 \).

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Key Concepts

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

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. In the given sequence, the common difference is 6, meaning each term is obtained by adding 6 to the previous term.

Recursive Formula

A recursive formula defines each term of a sequence based on the preceding term(s). In this case, the formula an = an-1 + 6 indicates that each term (an) is derived from the previous term (an-1) by adding 6. This approach is essential for generating terms in sequences where a direct formula is not provided.

Initial Term

The initial term of a sequence is the first term from which all subsequent terms are generated. In this problem, the initial term is given as a₁ = -9. This starting point is crucial for calculating the first six terms of the arithmetic sequence, as it sets the foundation for all following terms.