Problem 30 - Precalculus Mathematics for Calculus

Question

Find the nth term of a sequence whose first several terms are given. $$\frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \dots$$

Step-by-Step Solution

Answer

The nth term of the sequence is \( \frac{n+2}{n+3} \).

Step 1: Identify the Pattern

Observe the sequence given: \( \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \dots \). Notice that each term is a fraction where the numerator and the denominator increase by 1 consecutively. Specifically, each term is of the form \( \frac{n+2}{n+3} \), where \( n \) starts at 1 for the first term.

Step 2: Generalize the nth term

From our observation in Step 1, establish the general formula for the nth term. Since the numerators form the sequence 3, 4, 5, 6,..., the general term is \( n+2 \), and the denominators form the sequence 4, 5, 6, 7,..., the general term is \( n+3 \). Therefore, the nth term is \( \frac{n+2}{n+3} \).

Step 3: Verify the Formula

Verify the formula by substituting small values for \( n \). For \( n=1 \), the term is \( \frac{1+2}{1+3} = \frac{3}{4} \). For \( n=2 \), the term is \( \frac{2+2}{2+3} = \frac{4}{5} \), and for \( n=3 \), the term is \( \frac{3+2}{3+3} = \frac{5}{6} \). These calculations match the given terms in the sequence.

Key Concepts

Sequence Patterns

In mathematics, a sequence is an ordered list of numbers following a specific pattern or rule. Recognizing these patterns is crucial to identify the formula for the nth term. By carefully observing the given terms, we can uncover the rule that generates the rest of the sequence.

In our example sequence, the pattern is quite straightforward:

• The numerator of each term increases by 1, starting from 3.
• The denominator also increases by 1, starting from 4.

These observations allow us to deduce a rule: each term is of the form \( \frac{n+2}{n+3} \). By understanding and finding the sequence pattern, we create a formula for \( n \)-th term prediction.

Fraction Sequences

Fraction sequences involve terms that are fractions, and they often exhibit unique patterns involving both numerators and denominators. These sequences can be trickier to unravel because changes occur in two separate components.

In our task, every fraction moves incrementally:

• The numerator follows a simple pattern adding 1 to the previous term.
• The denominator progresses in a similar straightforward addition.

Recognizing these predictable fractional movements facilitates finding the general form for the fraction at any position in the sequence, resulting in the nth term \( \frac{n+2}{n+3} \). This systematic approach simplifies understanding and working with fraction sequences.

Mathematical Induction

Mathematical induction is a powerful method used to prove statements about an infinite number of elements, such as verifying the structure of sequences. It involves two main steps:

• Base Case: Verify the formula for the initial term.
• Inductive Step: Assuming the formula works for an arbitrary term, prove it holds for the next one.

To ensure our formula \( \frac{n+2}{n+3} \) is correct, we start by testing small values of \( n \), like 1, 2, and 3.

Each result confirms the proposed formula matches the given sequence. Induction guarantees that if it works for base cases and any general case \( n \, \) then it persists across all subsequent terms. It's a logical framework vital for validating solutions and maintaining consistency across mathematical sequences.