Finding a Sequence’s Formula
In Exercises 13–30, find a formula for the nth term of the sequence.
2, 6, 10, 14, 18, …Every other even positive integer

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Identify the pattern in the sequence: 2, 6, 10, 14, 18, ... Notice that each term increases by a constant difference of 4, which means this is an arithmetic sequence.

Recall the general formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference.

Substitute the known values into the formula: the first term \(a_1 = 2\) and the common difference \(d = 4\), so the formula becomes \(a_n = 2 + (n - 1) \times 4\).

Simplify the expression by distributing the 4: \(a_n = 2 + 4n - 4\).

Combine like terms to write the formula in simplest form: \(a_n = 4n - 2\).

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주요 개념

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Arithmetic Sequences

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference. Recognizing this pattern helps in formulating the nth term by identifying the first term and the common difference.

General Formula for the nth Term

The nth term of an arithmetic sequence can be expressed as a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. This formula allows direct calculation of any term in the sequence.

Sequence Pattern Recognition

Identifying the pattern in the given sequence, such as recognizing it lists every other even positive integer, is essential. This insight guides the choice of the first term and common difference in the formula.

Finding a Sequence’s FormulaIn Exercises 13–30, find a formula for the nth term of the sequence.2, 6, 10, 14, 18, …Every other even positive integer