91Ó°ÊÓ

In mathematics, the **nth-term** represents any term in a sequence based on its position, indicated by "n." This is the general term that allows you to calculate any member of the sequence without having to list all the terms that come before it. For example, let's consider the equation for a sequence given in this problem:
\[ a_n = \frac{(-1)^n}{n(n+1)} \] This formula expresses the nth-term of our sequence, with "n" representing the position of the term in the sequence. You can plug any integer value into "n" to find a specific term in the sequence. For our sequence:

• When \( n = 1 \), the first term \( a_1 \) evaluates to \( \frac{-1}{2} \).
• When \( n = 2 \), the second term \( a_2 \) is \( \frac{1}{6} \).
• Continues similarly for any given term position "n."

The formula simplifies computation and analysis of the sequence by providing a direct path to any term.

An **arithmetic sequence** is a sequence of numbers with the form where each term after the first is found by adding a constant, called the common difference, to the previous term. However, it's important to note that the sequence given in the problem is not an arithmetic sequence. An arithmetic sequence's common difference distinguishes it because it remains constant across all terms.
For example, if you had a sequence where the first term is 2 and the common difference is 3, the sequence would be: 2, 5, 8, 11, and so on.

• The common difference is 3, and it stays the same between each consecutive pair of terms.
• The general formula for an arithmetic sequence can be expressed as \( a_n = a_1 + (n-1) \cdot d \), where \( a_1 \) is the first term and \( d \) is the common difference.

While the sequence in the exercise uses an alternating sign, it does not fit the criteria for an arithmetic sequence because it lacks a constant difference between terms.

An **alternating sequence** is one where the terms regularly switch in sign, typically between positive and negative. This pattern is evident in the sequence given by the formula:
\[ a_n = \frac{(-1)^n}{n(n+1)} \]
The factor \((-1)^n\) is specifically influential in causing this alternation:

• When "n" is odd, \((-1)^n\) results in -1, giving the term a negative sign.
• When "n" is even, \((-1)^n\) results in 1, making the term positive.

Alternating sequences are important in many mathematical contexts, as they often help in calculations related to series, especially those involving convergence. The alternating nature can greatly affect the properties of a series derived from the sequence.