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A series in mathematics is the sum of the terms in a sequence. It provides a method to efficiently represent the addition of multiple numbers that follow a particular pattern. When you see a plus sign connecting numbers, like in this exercise, 1 + (-4) + 9 + (-16), they form a series. This specific type is known as a finite series because it only goes up to a certain number of terms.
You can express a series using summation notation, which simplifies the representation of long expressions. For example, the series we have here can be written as \( \sum_{n=1}^{4} (-1)^n \times n^2 \). The symbol \( \Sigma \) (sigma) is used to denote summation, and numbers above and below reveal the starting and stopping index. This notation helps mathematicians and students treat large sums in manageable segments.

A sequence is an ordered list of numbers that follow a specific rule or pattern. In the context of this exercise, the sequence is 1, -4, 9, -16. Each number in the sequence is called a term. Sequences can be finite, like this one, or infinite, extending indefinitely.
Understanding sequences is essential as they form the basis of series. The terms in our example are derived from a particular pattern involving squares and alternating signs. Recognizing these patterns is key when tackling problems related to sequences. Once you understand the sequence, you can easily transition to writing and working with the corresponding series.

A general term formula allows us to calculate any term in a sequence without listing all previous terms. This makes handling sequences and series much more efficient.
The given example uses the formula \((-1)^n \times n^2\). Here's how this works:

• \((-1)^n\) ensures the sign of the term alternates. When \(n\) is odd, \((-1)^n\) is -1, and when \(n\) is even, it's +1.
• \(n^2\) gives the square of the term number. For \(n = 1\), the term is \(1^2\); for \(n = 2\), it's \(2^2\), and so on.

This formula not only captures the alternating pattern of the sequence but also the fact that every term is a perfect square, making it easy to write and compute any term of the sequence.