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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 35
Chapter 9, Problem 35

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a7 when a1 = 2, r = 3

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Recall the formula for the nth term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
Identify the given values: \(a_1 = 2\), \(r = 3\), and \(n = 7\) (since we want to find \(a_7\)).
Substitute the known values into the formula: \(a_7 = 2 \times 3^{7-1}\).
Simplify the exponent: \(7 - 1 = 6\), so the expression becomes \(a_7 = 2 \times 3^6\).
Calculate \$3^6\( and then multiply by 2 to find the value of \)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 list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, if the first term is 2 and the ratio is 3, the sequence is 2, 6, 18, and so on.
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General Term Formula of a Geometric Sequence

The nth term of a geometric sequence can be found 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 direct calculation of any term without listing all previous terms.
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Exponentiation in Sequences

Exponentiation is used in the general term formula to raise the common ratio to the power of (n-1). Understanding how to compute powers is essential to correctly find terms in geometric sequences, such as calculating 3^(7-1) when finding the 7th term.
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