Problem 36 - Precalculus: Mathematics for Calculus

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

Step 1: Identify the Pattern

Observe the given sequence: \( \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \dots \). Notice that the numerator increases by 1 each time starting from 3, and the denominator simultaneously increases by 1 starting from 4. This suggests a pattern.

Step 2: General Formula Development

The numerators start at 3 and increase by 1 with each term, which can be expressed as \( n + 2 \). Similarly, the denominators start at 4 and also increase by 1 with each term, which can be expressed as \( n + 3 \). Therefore, the \( n \)th term can be generalized as \( \frac{n+2}{n+3} \).

Step 3: Construct the nth Term Equation

Combine the results from the previous step to construct the \( n \)th term equation for the sequence. The formula is \( a_n = \frac{n+2}{n+3} \).

Step 4: Verification

Substitute the first few values of \( n \) to verify if the pattern is accurately captured. For \( n = 1 \), \( a_1 = \frac{3}{4} \); for \( n = 2 \), \( a_2 = \frac{4}{5} \); for \( n = 3 \), \( a_3 = \frac{5}{6} \). This confirms that the derived formula is correct.

Key Concepts

nth term

The term 'nth term' in a sequence refers to a general expression that enables us to determine any term's value in the sequence without listing all the preceding terms. In this context, the nth term provides a formula for the sequence: \( a_n = \frac{n+2}{n+3} \). Here, \( n \) represents the position of the term within the sequence. This formula helps in finding any term of the sequence directly by plugging the position number into the formula. In our sequence, the nth term tells us how both the numerator and denominator evolve as the sequence progresses.

numerator

In fractions, the numerator is the top number that indicates how many parts of the whole are being considered. In the sequence \( \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \dots \), the numerators are 3, 4, 5, and 6, respectively. Recognizing the pattern of numerators is crucial for finding the nth term. In this sequence, each numerator increases by one from the previous term. To express this pattern mathematically for any nth term, we use \( n+2 \), starting with an initial value of 3 when \( n = 1 \).

denominator

The denominator, in a fraction, is the bottom number that represents the total number of equal parts. In the given sequence \( \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \ldots \), the denominators are 4, 5, 6, and 7. Just like the numerators, denominators increase by one as we move from one term to the next.
To represent the denominator for the nth term, the pattern is described by \( n+3 \). Thus, when \( n = 1 \), the denominator is 4, and the formula continues to hold for subsequent terms.

pattern recognition

Pattern recognition is a vital mathematical skill that involves identifying regularities, rules, or sequences in data. In this exercise, we applied pattern recognition to determine the formula for the nth term of a sequence. We observed that both the numerators and denominators increased by 1 as we moved from one term to the next.
This discovery guided us in expressing the sequence in its general form, \( \frac{n+2}{n+3} \). Recognizing such patterns helps in mathematics since it allows us to make predictions and generalizations about sequences or sets of numbers.