Problem 28 - Algebra and Trigonometry

Question

The given pattern continues. Write down the nth term of a sequence \(\left\\{a_{n}\right\\}\) suggested by the pattern. \(\frac{1}{1 \cdot 2}, \frac{1}{2 \cdot 3}, \frac{1}{3 \cdot 4}, \frac{1}{4 \cdot 5}, \ldots\)

Step-by-Step Solution

Answer

The nth term is \(a_{n} = \frac{1}{n \times (n+1)}\).

Step 1 - Observe the Pattern

Examine the initial terms to identify the pattern: \(\frac{1}{1 \times 2}, \frac{1}{2 \times 3}, \frac{1}{3 \times 4}, \frac{1}{4 \times 5}, \rightarrow \frac{1}{n \times (n+1)}\).

Step 2 - Generalize the Formula

From the observations, note that each term can be written in the form \(\frac{1}{n \times (n+1)}\), where \(n\) starts from 1 and increases by 1 for each subsequent term.

Step 3 - Write the nth Term

Based on the pattern observed, the nth term of the sequence is: \(a_{n} = \frac{1}{n \times (n+1)}\).

Key Concepts

Sequences and Series

In mathematics, a sequence is an ordered list of numbers. Each number in a sequence is called a term. For example, in the sequence given in the exercise - \(\frac{1}{1 \times 2}, \frac{1}{2 \times 3}, \frac{1}{3 \times 4}, \frac{1}{4 \times 5}, \) - each fraction represents a term.
A series, on the other hand, is the sum of the terms of a sequence. In this case, we are only dealing with a sequence, not a series.
Understanding the difference between the two is important. A sequence lists numbers in a specific order, while a series adds them up.

Pattern Recognition

The ability to recognize patterns is crucial in solving problems involving sequences. In this exercise, we are tasked with identifying how each term in the sequence relates to its position (n).
Start by examining the initial terms:

• First term: \(\frac{1}{1 \times 2}\)
• Second term: \(\frac{1}{2 \times 3}\)
• Third term: \(\frac{1}{3 \times 4}\)
• And so on.

Notice that the denominator in each term can be written as \(n \times (n+1)\). Recognizing this consistent pattern is key to formulating the nth term.

General Term Formula

Once we have identified the pattern, the next step is to generalize the formula for the nth term. From our observation:

• For n=1, the term is \(\frac{1}{1 \times 2}\)
• For n=2, the term is \(\frac{1}{2 \times 3}\)
• For n=3, the term is \(\frac{1}{3 \times 4}\)

We see that each term follows a specific rule: the numerator is always 1, and the denominator is the product of n and \(n+1\). Hence, we can write the nth term as:
\[a_{n} = \frac{1}{n \times (n+1)}\]

Algebraic Sequences

An algebraic sequence is a sequence where each term is derived using a specific algebraic formula. Algebraic sequences often have a clear pattern and mathematical relationship between the terms.
In our given exercise, the sequence \(\frac{1}{1 \times 2}, \frac{1}{2 \times 3}, \frac{1}{3 \times 4}, \frac{1}{4 \times 5}, \) is an excellent example of an algebraic sequence. Here, each term is expressed as an algebraic fraction.
Recognizing algebraic sequences and understanding their formulas can simplify the process of finding any term within the sequence. It transforms complex ideas into manageable calculations and allows for general understanding beyond just memorizing terms.