Problem 14 - Precalculus with Limits

Question

In Exercises 5 - 16, determine whether the sequence is geometric. If so, find the common ratio. \( \dfrac{1}{5}, \dfrac{2}{7}, \dfrac{3}{9}, \dfrac{4}{11}, \cdots \)

Step-by-Step Solution

Answer

The given sequence is not a geometric sequence as the ratio between consecutive terms is not constant.

Step 1: Compute ratio for the first two terms

Calculate the ratio of the second term to the first term: \( r_1 = \frac{\dfrac{2}{7}}{\dfrac{1}{5}} = \frac{2}{7} \times \frac{5}{1} = \frac{10}{7} = 1.\overline{428571} \).

Step 2: Compute ratio for the second and third terms

Calculate the ratio of the third term to the second term: \( r_2 = \frac{\dfrac{3}{9}}{\dfrac{2}{7}} = \frac{3}{9} \times \frac{7}{2} = \frac{21}{18} = 1.\overline{16} \) .

Step 3: Compare the ratios

Compare \( r_1 \) and \( r_2 \). If they are equal, sequence is geometric and the common ratio is \( r_1 \) or \( r_2 \). However, here, they are unequal since \( 1.\overline{428571} \ne 1.\overline{16} \). This means the sequence is not geometric.

Key Concepts

Common Ratio

The term **common ratio** is essential when discussing geometric sequences. A geometric sequence is a list of numbers where each term is derived from the previous one by multiplying it with a fixed number, known as the common ratio. To determine if a sequence is geometric, you must check if the ratios between successive terms are constant.

For the sequence given in the exercise, which is \( \frac{1}{5}, \frac{2}{7}, \frac{3}{9}, \frac{4}{11}, \cdots \), you find the common ratio by dividing each term by the previous term. If all these ratios are the same, you have a geometric sequence, and that number is your common ratio. But if they're not the same, like in this exercise, the sequence is not geometric.

• Example: Calculate the common ratio between the first two terms \( \frac{2}{7} \) divided by \( \frac{1}{5} \).
• If you obtain a different ratio between any pair of successive terms, there is no common ratio.

Mathematical Sequence

Understanding what a **mathematical sequence** is can significantly help in solving sequence-related problems. A sequence is an ordered set of numbers following a specific rule, and can be finite or infinite.

In a sequence, each number is called a term. There are different types of sequences, but the most common are arithmetic and geometric sequences. In an arithmetic sequence, each term is the addition of a fixed number to the previous term. Conversely, a geometric sequence uses multiplication with a fixed ratio, known as the common ratio, to achieve subsequent terms.

• Arithmetic Sequence Example: 2, 4, 6, 8 (difference is 2)
• Geometric Sequence Example: 3, 6, 12, 24 (common ratio is 2)

Thus, to categorize a given sequence as geometric, consistency of the common ratio is essential.

Ratio Calculation

**Ratio calculation** is a simple yet highly integral process when working with sequences. It's especially important in identifying geometric sequences. To calculate a ratio, dividing one magnitude by another gives you a figure that represents how many times the second term fits into the first.

For sequence problems like the one presented, you calculate the ratio between consecutive terms to determine if it's geometric. For example, for the sequence provided:

• First calculate: \( r_1 = \frac{\left( \frac{2}{7} \right)}{\left( \frac{1}{5} \right)} = \frac{10}{7} \)
• Then calculate: \( r_2 = \frac{\left( \frac{3}{9} \right)}{\left( \frac{2}{7} \right)} = \frac{21}{18} \)
• Comparing \( r_1 \) and \( r_2 \), they must be equal for the sequence to be geometric.

In situations where these calculations yield different results, as shown in this exercise, the sequence cannot be geometrically classified.