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Sequences are ordered lists of numbers following a specific pattern or rule. In sequence notation, the general term of the sequence is expressed as a function of its position number, usually denoted by the variable \( n \). For example, the sequence you are working with is represented as \( a_n = \frac{(-1)^n}{(n+1)(n+2)} \). This formula is the blueprint for generating the sequence's terms.

A sequence can begin at any value of \( n \), but it's common for it to start at \( n = 1 \). Each distinct \( n \) creates a different term in the sequence. When reading or writing sequence notation, it’s important to remember:

• \( a_n \) refers to the \( n \)-th term of the sequence.
• The formula for \( a_n \) determines the rule that every term follows.
• Evaluating the sequence at specific values of \( n \) gives the sequence's terms.

This notation allows mathematicians to succinctly express potentially infinite series of numbers, showing both the position and value of each term in a concise manner.

To calculate terms in a sequence, substitute specific values of \( n \) into the formula. This process extracts specific terms from the overall pattern defined by the sequence.

Consider the sequence \( a_n = \frac{(-1)^n}{(n+1)(n+2)} \). To find the first term, replace \( n \) with 1:
\[ a_1 = \frac{(-1)^1}{(1+1)(1+2)} = \frac{-1}{6} \].
This calculation yields the first term, \(-\frac{1}{6}\).

Continue this process for subsequent terms:

• Second term: \( a_2 = \frac{1}{12} \)
• Third term: \( a_3 = \frac{-1}{20} \)
• Fourth term: \( a_4 = \frac{1}{30} \)

Remember to perform each calculation step-by-step. Substitute the value of \( n \) correctly, compute powers of \(-1\) carefully, and simplify the fractions, which helps ensure accuracy as you build the sequence.

The sequence in this exercise is an example of an alternating series. An alternating series is a sequence of numbers where consecutive terms change sign. The sign changes are integral to the function governing the sequence, often facilitated by powers of \(-1\).

In the sequence \( a_n = \frac{(-1)^n}{(n+1)(n+2)} \):

• \((-1)^n\) provides the alternating sign, because when \( n \) is odd, \((-1)^n = -1\), and when \( n \) is even, \((-1)^n = 1\).
• This alternation causes the sequence to switch between negative and positive values as \( n \) increases.

This characteristic of alternating series affects calculations and interpretations. It informs us that as we evaluate further terms, each will inherently change sign relative to its predecessor, creating a back-and-forth pattern. Such patterns are often used in approximations and can converge to particular values, making them valuable in mathematical analysis.