Problem 17 - Precalculus Mathematics for Calculus

Question

Determine whether the sequence is geometric. If it is geometric, find the common ratio. $$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots$$

Step-by-Step Solution

Answer

The sequence is not geometric as it lacks a constant common ratio.

Step 1: Identify the Terms

Let's list out the terms of the sequence given:1st term: \( \frac{1}{2} \)2nd term: \( \frac{1}{3} \)3rd term: \( \frac{1}{4} \)4th term: \( \frac{1}{5} \)Each term is given in the form of a fraction, where the numerator is always 1, and the denominator is consecutive integers starting from 2.

Step 2: Check for a Common Ratio

A sequence is geometric if the ratio between consecutive terms is constant. We calculate this ratio for the terms:Ratio between 1st and 2nd term:\[ r_1 = \frac{1/3}{1/2} = \frac{1}{3} \times \frac{2}{1} = \frac{2}{3} \]Ratio between 2nd and 3rd term:\[ r_2 = \frac{1/4}{1/3} = \frac{1}{4} \times \frac{3}{1} = \frac{3}{4} \]Ratio between 3rd and 4th term:\[ r_3 = \frac{1/5}{1/4} = \frac{1}{5} \times \frac{4}{1} = \frac{4}{5} \]

Step 3: Analyze the Ratios

As we can see, each ratio between consecutive terms is different:\( r_1 = \frac{2}{3} \)\( r_2 = \frac{3}{4} \)\( r_3 = \frac{4}{5} \)Therefore, the sequence does not have a constant ratio between terms.

Step 4: Conclusion

Since the ratio between consecutive terms is not constant, the sequence is not geometric. A geometric sequence requires a constant ratio between successive terms.

Key Concepts

Common Ratio

A common ratio is a key feature in understanding geometric sequences. It refers to the constant number by which each term in a geometric sequence is multiplied to get the next term.
In a formulaic sense, if you have a sequence where each subsequent term can be obtained by multiplying the previous term by a fixed number, this number is called the common ratio, denoted by \( r \).

The common ratio can be calculated by dividing any term by the previous term in the sequence. For example, if you have terms \( a_1 \), \( a_2 \), \( a_3 \) of a sequence, the common ratio \( r \) can be found as follows:

• \( r = \frac{a_2}{a_1} \)
• \( r = \frac{a_3}{a_2} \)

In this exercise, the terms \( \frac{1}{2} \), \( \frac{1}{3} \), \( \frac{1}{4} \), and \( \frac{1}{5} \) were tested to see if such a common ratio existed.
Since the calculations showed different ratios between each pair of terms, we concluded that no common ratio exists, indicating that this sequence is not geometric.

Sequence Analysis

Sequence analysis involves examining a series of numbers to determine a pattern or rule that defines the sequence.
There are different types of sequences, like arithmetic and geometric, that follow specific rules. Understanding these rules helps us categorize sequences and predict future terms.

In this exercise, the task was to determine if the given sequence was geometric:

• We studied the pattern of the numerators and denominators of the terms. Here, each numerator was \( 1 \), while the denominators were consecutive integers starting from 2.
• By calculating the ratios of consecutive terms, we looked for a constant ratio. However, the differing ratios \( \frac{2}{3} \), \( \frac{3}{4} \), and \( \frac{4}{5} \) indicated that the sequence was not geometric.

The lack of a constant ratio is crucial to identify, as it directly affects the classification of the sequence, distinguishing it from a geometric sequence.

Mathematical Fractions

Understanding fractions is important when dealing with sequences, especially when they involve terms expressed as fractions.
A fraction consists of a numerator (the top number) and a denominator (the bottom number).
In the sequence \( \frac{1}{2} \), \( \frac{1}{3} \), \( \frac{1}{4} \), \( \frac{1}{5} \), the numerators are all \( 1 \), indicating equal fractional parts.

To analyze these fractions within the sequence, we looked at their ratios:

• The ratio of two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) is calculated as \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \).
• This calculation helps us to compare relative sizes of fractions within the sequence, as seen when finding the ratios \( \frac{2}{3} \), \( \frac{3}{4} \), and \( \frac{4}{5} \).

By mastering the manipulation of fractions, students can better understand the makeup of various kinds of sequences and determine their characteristics more effectively.