Problem 28 - Algebra and Trigonometry

Question

Find the \(n\)th 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 is \(a_n = \frac{n+2}{n+3}\).

Step 1: Observe the Pattern

The sequence given is \( \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \dots \). Notice that each term is a fraction where the numerator starts at 3 and increases by 1 for each subsequent term, and the denominator starts at 4 and also increases by 1 for each subsequent term.

Step 2: Define the General Term

Based on the observation from Step 1, you can discern that the numerator of the sequence's terms increases by 1 sequentially: 3, 4, 5, 6, which can be expressed as \(n + 2\). Similarly, the denominators are 4, 5, 6, 7, which can be expressed as \(n + 3\). This pattern suggests that the general term of the sequence \(a_n\) can be defined by the formula \(a_n = \frac{n+2}{n+3}\).

Step 3: Test the Formula

To ensure the formula \(a_n = \frac{n+2}{n+3}\) is correct, substitute the initial terms' positions (1, 2, 3, etc.) into \(n\) and verify: - For \(n=1\), \(a_1 = \frac{1+2}{1+3} = \frac{3}{4}\).- For \(n=2\), \(a_2 = \frac{2+2}{2+3} = \frac{4}{5}\).- For \(n=3\), \(a_3 = \frac{3+2}{3+3} = \frac{5}{6}\).Each calculation confirms the observed sequence.

Key Concepts

Understanding Sequence Patterns

A sequence is a set of numbers arranged in a specific order. Recognizing the pattern within a sequence is key to identifying its general term. This particular sequence consists of fractions: \( \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \dots \). To spot the pattern, examine how both the numerators and denominators change as you move from one term to the next.

• The numerator starts at 3 and increases by 1: 3, 4, 5, 6 ...
• Similarly, the denominator starts at 4 and also increases by 1: 4, 5, 6, 7 ...

Recognizing these patterns helps us determine the rule that generates the sequence. Remember, the goal is to identify how each part of the term changes in relation to the term's position \( n \) in the sequence.

Creating the General Term Formula

Once the pattern in the sequence is understood, we can define the general term formula. This formula allows us to find any term in the sequence without listing all the preceding ones. Based on our observed patterns,
the numerators are formed by the expression \( n + 2 \) (since the sequence starts at 3 when \( n = 1 \)) and the denominators by \( n + 3 \) (since the sequence starts at 4 when \( n = 1 \)).
Hence, the general term \( a_n \) of the sequence can be expressed as:\[ a_n = \frac{n + 2}{n + 3} \]This formula captures the essence of the sequence by describing the relationship between a term's position and its value. Always verify this formula by substituting initial values to ensure it replicates the given terms correctly.

Exploring Fractional Sequences

Fractional sequences consist of terms that are fractions, where both the numerator and denominator follow a distinct pattern. Unlike whole number sequences, fractional sequences can often be trickier, because they involve two changing components: the numerator and the denominator.
To tackle such sequences effectively, it's crucial to:

• Identify how shifts in position \( n \) affect both parts of the fraction.
• Look for systematic increases or decreases in both the numerator and denominator.

In our exercise, each fraction \( \frac{n+2}{n+3} \) is formed by adding constants to \( n \), maintaining a straightforward relationship between terms. Mastery of fractional sequences requires practice in deciphering these small yet consistent patterns.