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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1:35 minutes

Problem 35

Textbook Question

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 7) when a(sub 1) = 2, r = 3

Verified step by step guidance

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

Substitute the given values into the formula: \( a_1 = 2 \), \( r = 3 \), and \( n = 7 \).

The formula becomes \( a_7 = 2 \cdot 3^{(7-1)} \).

Simplify the exponent: \( 7 - 1 = 6 \), so the expression is \( a_7 = 2 \cdot 3^6 \).

Calculate \( 3^6 \) and then multiply the result by 2 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 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. For example, in the sequence 2, 6, 18, 54, the common ratio is 3, as each term is obtained by multiplying the previous term by 3.

General Term of a Geometric Sequence

The general term (nth term) of a geometric sequence can be expressed using the formula a(n) = a(1) * r^(n-1), where a(1) is the first term, r is the common ratio, and n is the term number. This formula allows us to calculate any term in the sequence based on its position and the initial values.

Substitution in Formulas

Substitution in formulas involves replacing variables with their corresponding values to compute a specific result. In this context, to find a(sub 7), we substitute a(1) = 2 and r = 3 into the general term formula, allowing us to calculate the seventh term of the sequence accurately.