The first few terms of a sequence are given. Find an...

The first few terms of a sequence are given. Find an expression for the nth term of each sequence, assuming the indicated pattern continues for all $n$.
$$
\frac{1}{2},-\frac{1}{3}, \frac{1}{4},-\frac{1}{5}, \frac{1}{6}, \ldots
$$

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Key Concepts

Alternating Sequences

An alternating sequence is one in which the terms switch signs, usually in a regular pattern such as positive, negative, positive, and so on. The use of factors like (-1)^n or (-1)^(n+1) is typical to represent this alternation mathematically, and understanding this concept helps in dealing with series that converge under specific conditions.

Pattern Recognition in Sequences

This concept involves identifying regular trends or patterns in the terms of a sequence, such as consistent changes in the numerator or denominator, or predictable sign changes. Recognizing these patterns is crucial for deriving a general formula for the nth term, which can then be applied universally across the sequence.

General Term (nth Term) Expression

This concept involves formulating an expression that can directly calculate the nth term of a sequence without needing to list all previous terms. It is fundamental in the study of sequences and series as it provides a concise way to describe the behavior of the sequence for any given positive integer n.

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