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

Geometric Sequences

College Algebra

9. Sequences, Series, & Induction

Geometric Sequences

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2:04 minutes

Problem 90

Textbook Question

Exercises 88–90 will help you prepare for the material covered in the next section. Use the formula an = a₁3^(n-1) to find the seventh term of the sequence 11, 33, 99, 297,...

Verified step by step guidance

Identify the first term $ a_1 $ of the sequence, which is given as 11.

Recognize that the sequence follows a geometric pattern with a common ratio. Calculate the common ratio $ r $ by dividing the second term by the first term: $ r = \frac{33}{11} = 3 $.

Use the formula for the nth term of a geometric sequence: $ a_n = a_1 \cdot r^{(n-1)} $.

Substitute the known values into the formula to find the seventh term: $ a_7 = 11 \cdot 3^{(7-1)} $.

Simplify the expression by calculating $ 3^6 $ and then multiplying the result by 11 to find $ a_7 $.

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Key Concepts

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

Geometric Sequences

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. In this case, the sequence 11, 33, 99, 297 has a common ratio of 3, as each term is three times the previous one.

Explicit Formula for Geometric Sequences

The explicit formula for a geometric sequence is given by an = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number. This formula allows you to calculate any term in the sequence directly without needing to find all preceding terms.

Finding Specific Terms

To find a specific term in a sequence using the explicit formula, substitute the values of a₁, r, and n into the formula. For example, to find the seventh term of the sequence, you would set a₁ = 11, r = 3, and n = 7, and then calculate a₇ = 11 * 3^(7-1).