Sequences are fundamental mathematical concepts used to model and predict patterns both in theoretical contexts and real-world applications. Essentially, a sequence is a function where the domain consists of positive integers starting from one, two, three, and so forth. Each input, called an index (denoted by n), corresponds to an output known as a term, represented as a_n. This notation allows us to express sequences as ordered lists of numbers, where a₁ is the first term, a₂ the second, and so on.

Understanding sequences involves evaluating the general term formula by substituting different index values. For example, if the general term is given by a_n = 2ⁿ, the first term is found by plugging in n = 1, yielding a₁ = 2¹ = 2. Similarly, the second term is a₂ = 2² = 4, and the third term is a₃ = 2³ = 8. This process can be continued to find any term in the sequence, such as the tenth term, calculated as a₁₀ = 2¹⁰ = 1024. Representing sequences as lists, such as 2, 4, 8, ..., helps visualize the pattern and understand its progression.

Sequences can be classified as either infinite or finite. An infinite sequence continues indefinitely without end, as seen in the example above where the index n can take any positive integer value. In contrast, a finite sequence has a restricted domain, such as when n is limited to values between 1 and 5, resulting in a sequence that stops after a certain number of terms. Recognizing these distinctions is crucial for analyzing and working with sequences in various mathematical contexts.

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