Problem 27 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Numerator Pattern

Observe the numerical pattern of the sequence: \ \ \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots \). The numerators are 1, 2, 3, 4, ... and so on. Thus, the numerator for the nth term is simply n.

Step 2: Identify the Denominator Pattern

Next, observe the denominators of the sequence: \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots \). The denominators are 2, 3, 4, 5, ... which are n + 1 for each corresponding numerator. Therefore, the denominator for the nth term is n + 1.

Step 3: Combine the Patterns

Combine the patterns identified for the numerator and the denominator. The nth term of the sequence can be expressed as: \ \ \(a_{n} = \frac{n}{n+1}\)

Step 4: Write the General Term

Therefore, the general term for the sequence is: \ \ \(a_{n} = \frac{n}{n+1}\)

Key Concepts

Numerator Pattern

Identifying the numerator pattern in a sequence is crucial for understanding how to find the nth term. When looking at the sequence \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\), observing the numerators is the first step.

The numerators in this sequence are clearly following a simple arithmetic pattern: 1, 2, 3, 4,\ldots. Notice how each term increases by 1 compared to the previous term. This consistent increase by 1 implies that the numerator for the nth term is simply n.

So, for our sequence, the numerator of the nth term can be denoted as \(n\).

Denominator Pattern

Next, we need to identify the pattern in the denominators, which can sometimes be a bit trickier. Examine the given sequence again: \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\).

We see another clear pattern: the denominators are 2, 3, 4, 5, \ldots. To find a relation to the numerators, compare each numerator and its corresponding denominator:

• The denominator of the first term is 2 (1 + 1).
• The denominator of the second term is 3 (2 + 1).
• The denominator of the third term is 4 (3 + 1).

The denominators are always one more than their respective numerators. Thus, the denominator for the nth term in the sequence can be expressed as \(n + 1\).

General Term Formula

After identifying the patterns in both the numerators and the denominators, we can combine these to find the general term formula.

From our observations:

• The numerator of the nth term is \(n\).
• The denominator of the nth term is \(n + 1\).

Therefore, the nth term of the sequence can be written as \(a_{n} = \frac{n}{n+1}\).

This formula is crucial as it provides a simple and concise method to find any term in the sequence, without having to list all the preceding terms. To reinforce understanding, you can plug in different values for n and watch the pattern hold:

• For \(n = 1\), \(a_{1} = \frac{1}{2}\).
• For \(n = 2\), \(a_{2} = \frac{2}{3}\).
• For \(n = 3\), \(a_{3} = \frac{3}{4}\).

Understanding these patterns and how they combine into the general term formula allows a deeper grasp of sequence behavior.