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Sequences and series are fundamental concepts in arithmetic that help us understand ordered lists of numbers. A **sequence** is an ordered collection of terms that follows a specific pattern. Each element of a sequence is known as a term. Sequences can be finite, with a limited number of terms, or infinite, continuing indefinitely.
A **series** is the sum of the terms of a sequence. It offers a way to sum up an infinite sequence of numbers. For example, the series of a finite arithmetic sequence with terms 1, 2, and 3 would be 1 + 2 + 3 = 6.
In this exercise, we dealt with a sequence defined by the formula \(a_n = \frac{n}{n+1}\), which is neither an arithmetic nor a geometric sequence. Understanding sequences and series can help us explore different types of numerical patterns and solve real-world problems that involve them.

Each term in a sequence is shaped by a formula, allowing us to determine any term without needing to write down all the previous terms. This exercise's sequence is given by \(a_n = \frac{n}{n+1}\). To find a specific term, one substitutes the term number \(n\) into this formula.
For instance:

• The first term \(a_1\) is found by substituting \(n = 1\), resulting in \(a_1 = \frac{1}{2}\).
• The second term \(a_2\) is obtained by using \(n = 2\), which gives \(a_2 = \frac{2}{3}\).
• Continuing this pattern, the third term \(a_3\) becomes \(\frac{3}{4}\), and so on.

Understanding how to identify terms of a sequence helps in exploring the overall behavior of the sequence and enables one to analyze the relationships between consecutive terms.

Fractional expressions are widely used in mathematics to represent numbers that are not whole. A **fraction** is comprised of a numerator (the top number) and a denominator (the bottom number).
In the sequence \(a_n = \frac{n}{n+1}\), each term is a fractional expression.

• The numerator \(n\) indicates the term number.
• The denominator \(n+1\) is one more than the term number.

This results in a fraction that varies as \(n\) increases.
The importance of fractional expressions lies in their ability to express quantities between whole numbers. They are essential in both pure and applied mathematics, helping to resolve problems involving ratios, divisions, and proportional relationships.