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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 31
Chapter 9, Problem 31

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. an = an-1 +3, a1 = 4

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Identify the first term of the arithmetic sequence, which is given as \(a_1 = 4\).
Recognize that the sequence increases by a common difference of 3 each time, as indicated by the recursive formula \(a_n = a_{n-1} + 3\).
Write the explicit formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1)d\), where \(d\) is the common difference.
Substitute the known values into the formula: \(a_n = 4 + (n - 1) \times 3\).
To find the 20th term, substitute \(n = 20\) into the explicit formula: \(a_{20} = 4 + (20 - 1) \times 3\).

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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 list of numbers where each term after the first is found by adding a constant difference to the previous term. This constant difference is called the common difference. For example, in the sequence 4, 7, 10, 13, ..., the common difference is 3.
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General Term Formula of an Arithmetic Sequence

The general term (nth term) of an arithmetic sequence can be expressed as an = a₁ + (n - 1)d, where a₁ is the first term, d is the common difference, and n is the term number. This formula allows direct calculation of any term without recursion.
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Recursive vs Explicit Formulas

A recursive formula defines each term based on the previous term(s), while an explicit formula gives a direct expression for the nth term. Converting a recursive formula like an = an-1 + 3 into an explicit formula simplifies finding specific terms, such as the 20th term.
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