Each of Exercises 1-6 gives a formula for the n th term an...

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a
sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4} .$
$$
a_{n}=\frac{(-1)^{n+1}}{2 n-1}
$$

Key Concepts

Sequence

A sequence is an ordered list of numbers where each number is called a term. In mathematics, sequences are typically described by a formula that assigns a value to each term based on its position in the sequence, and they serve as a foundation for many topics including series, limits, and functions.

Nth Term Formula

The nth term formula provides a method to determine any term in a sequence directly based on its position. This formula is a crucial tool for identifying specific elements in the sequence without computing all the previous terms, thereby offering efficiency and clarity in understanding how the sequence behaves.

Substitution and Evaluation

Substitution involves replacing the variable in an expression or formula with a specific value, allowing for the evaluation of a term or function at that value. In the context of sequences, this means plugging in the term's position to calculate its corresponding value, which is essential for understanding the structure and behavior of sequences.

Alternating Signs in Sequences

Alternating sign sequences incorporate a factor, often expressed as (-1)^n or a similar variant, to produce a pattern of positive and negative values. This technique is a common method to introduce sign variation in sequences and series, influencing convergence properties and the overall behavior of the sequence.

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