Problem 7 - Single Variable Calculus

Question

Find an expression for the \(n\) th term of the sequence. (Assume that the pattern continues.) \(\left\\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \ldots\right\\}\)

Step-by-Step Solution

Answer

The short answer for the nth term of the given sequence is \(a_n = \frac{n}{n+1}\).

Step 1: Identify the pattern

Looking closely at the sequence, we can notice that the numerator of each term is 1 less than the denominator. For example, in the first term, the numerator is 1 and the denominator is 2, then for the second term, the numerator is 2 and the denominator is 3, so on and so forth.

Step 2: Generate a general expression for the sequence

Now that we have observed a pattern in the way the sequence is formed, we can generate a general expression as a function of the term number, 'n'. Since the numerator is always 1 less than the denominator, the nth term will have a numerator of 'n' and a denominator of 'n+1', and hence the expression for the nth term of the sequence is as follows: \[ a_n = \frac{n}{n+1} \] The given expression, \(a_n = \frac{n}{n+1}\), is the nth term of the sequence and can be used to find any term in the sequence by simply substituting the value of 'n'.

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a list of numbers with a common difference between consecutive terms. This difference is constant across all the terms in the sequence. It allows us to predict and find any term when we know the previous terms.

In general, an arithmetic sequence can be expressed with the formula:

• First term, denoted as \(a_1\), which is the starting point.
• Common difference, denoted as \(d\), which is added to each term to get the next one.

Thus, the sequence is defined as \(a_n = a_1 + (n-1)d\).

In our exercise, however, the sequence given is not arithmetic because the pattern does not include adding or subtracting a constant difference. Instead, it involves a relationship between the numerator and denominator of each fraction.

Pattern Recognition

Pattern recognition is a crucial skill in identifying and understanding sequences. It involves looking at the sequence and spotting trends or relationships between terms.

In our given sequence \[\left\{ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \ldots \right\}\],
we observe a specific pattern:

• Each numerator increases by 1, starting from 1, 2, 3, ..., which aligns with the term number \(n\).
• Each denominator is 1 more than its corresponding numerator.

Therefore, the pattern reveals a relationship where the numerator of the nth term is \(n\), and the denominator is \(n+1\). Understanding and applying this pattern enables us to predict every term in the sequence without calculating them one by one.

General Term Formula

The general term formula is an equation that provides a way to calculate any term in a sequence without listing all preceding terms. It encapsulates the pattern of the sequence into a single expression.

For the given sequence \(\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \ldots\right\}\),
once we recognize the pattern, we can establish the general term formula as:

• Denote the nth term as \(a_n\).
• Observe from the pattern that \(a_n = \frac{n}{n+1}\).

This formula \(a_n = \frac{n}{n+1}\) allows us to find any term in the sequence simply by plugging in the desired term number \(n\). Whether you're looking for the fifth, tenth, or fiftieth term, this formula provides a direct computation method.