The first several terms of a sequence a1, a2 ⋯are given....

The first several terms of a sequence $a_{1}, a_{2} \cdots$ are given. Assume that the pattern continues as indicated and find an explicit formula for the $a_{n}$
$$1,2, \frac{1}{9}, 4, \frac{1}{25}, 6, \frac{1}{49}, \dots,$$

Key Concepts

Parity Consideration

Parity refers to the classification of integers as either even or odd. Many problems and sequences exhibit different patterns or formulas based on whether the index is even or odd, making parity an important tool for constructing piecewise definitions.

Subsequences

Subsequences are formed by taking elements from the original sequence that meet certain criteria, such as all even-indexed terms or all odd-indexed terms. Analyzing these subsequences can help uncover hidden patterns and simplify the process of finding an explicit formula for the overall sequence.

Piecewise Function

A piecewise function is one that is defined by different expressions over different parts of its domain. In the context of sequences, this concept is often used when the sequence exhibits different behaviors based on properties such as the index being odd or even.

Sequence

A sequence is an ordered collection of elements or numbers defined by a specific rule or pattern. It is a fundamental concept in mathematics that helps in understanding structures, patterns, and behaviors within a set of numbers.

Explicit Formula

An explicit formula provides a direct expression to compute the nth term of a sequence solely in terms of n, without needing the previous terms. This is useful for analyzing the properties of the sequence and for efficiently finding any term in the sequence.

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