Question

Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) \(\frac{1}{2 \cdot 3}, \frac{2}{3 \cdot 4}, \frac{3}{4 \cdot 5}, \frac{4}{5 \cdot 6}, \ldots\)

Short answer

The \(n\) th term of the sequence is given by the expression \(\frac{n}{(n+1)(n+2)}\).

Step-by-step solution

Identify the Pattern

Look at the sequence: \(\frac{1}{2 \cdot 3}, \frac{2}{3 \cdot 4}, \frac{3}{4 \cdot 5}, \frac{4}{5 \cdot 6}, \ldots\). Observe that the numerator of each fraction is increasing by 1, and the denominator of each fraction involves multiplying two consecutive integers which are also incrementing by 1 in each term.

Formulate the \(n\) th term

Based on the pattern observed, we can write the \(n\) th term of the sequence as: \(\frac{n}{(n+1)(n+2)}\). The numerator is \(n\), and the denominator is the product of the next two integers after \(n\), which is \((n+1)(n+2)\). Just insert \(n\) for every term.

Key concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a common difference between consecutive terms. It's quite simple to spot: if you subtract any term from the next one, you'll always get the same result. For example, in the sequence 3, 7, 11, 15, ..., the common difference is 4.

In the provided exercise, the sequence does not have a constant difference, indicating it's not a standard arithmetic sequence. Each term is represented by a fraction, and the difference does not remain constant. However, understanding arithmetic sequences helps in grasping more complex sequences, like the one in the exercise, because it sharpens one's ability to look for patterns. The takeaway here is pattern recognition, which is fundamental in all types of sequences.

Fraction Sequences

Working with fractions in sequences can be a bit tricky at first glance, but if you understand the underlying pattern, it's no different from working with whole numbers. Fraction sequences, like the one in the exercise, often involve a pattern in both the numerators and denominators.

To approach these, break down the pattern in each part of the fraction. Look for increments or other relationships in the numerators and denominators separately. As seen in the example, the numerator increases by 1 with each term, and the denominator is the product of consecutive integers, which also presents a clear pattern when looked at closely. Recognizing such patterns is crucial in forming an expression for the nth term.

Sequence Patterns

Identifying sequence patterns is like solving a puzzle. You look for regularities and relationships between the terms. There are many types of patterns, including arithmetic and geometric sequences, or even more complex formulas involving squares, cubes, or variable denominators as in fractional sequences.

In our exercise, the sequence pattern involves increasing numerators and a specific relationship in the denominators that can be described by consecutive integers. Understanding different types of sequence patterns is essential for finding the nth term because it tells you how to generalize the sequence's formula effectively. Practice with various patterns strengthens your ability to discern and describe these relationships mathematically.

Calculus

While our sequence does not directly involve calculus, this field of mathematics is all about studying change and motion, often dealing with continuous sequences and series. In calculus, sequences appear as discrete counterparts to continuous functions, and understanding the behavior of sequences is key to grasping concepts like limits, derivatives, and integrals, which are the building blocks of calculus.

Although you don't need calculus to solve our current exercise, the concept of recognizing patterns and describing them using functions is relevant. For more complex sequences that may not have a straightforward pattern, calculus provides tools, like summation formulas and series convergence tests, that allow us to understand and sum infinite sequences.