Problem 19 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Determine whether these terms can be the terms of a geometric sequence. If the sequence 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 there is no constant common ratio.

Step 1: Identify the Given Terms

The first four terms of the sequence are given as: \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \).

Step 2: Recall the Definition of a Geometric Sequence

A geometric sequence is one where each term after the first is found by multiplying the previous term by the same number, called the common ratio \( r \). Mathematically, for a sequence to be geometric, \( \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = r \).

Step 3: Calculate the Ratio Between Terms

We need to check if there is a constant common ratio. Calculate the ratios between consecutive terms:1st ratio: \( \frac{a_2}{a_1} = \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{1}{3} \times \frac{2}{1} = \frac{2}{3} \),2nd ratio: \( \frac{a_3}{a_2} = \frac{\frac{1}{4}}{\frac{1}{3}} = \frac{1}{4} \times \frac{3}{1} = \frac{3}{4} \),3rd ratio: \( \frac{a_4}{a_3} = \frac{\frac{1}{5}}{\frac{1}{4}} = \frac{1}{5} \times \frac{4}{1} = \frac{4}{5} \).

Step 4: Assess the Constancy of Ratios

Examine the calculated ratios: \( \frac{2}{3}, \frac{3}{4}, \frac{4}{5} \). Since these ratios are not equal, the sequence does not have a constant common ratio.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is an essential component. It is the consistent factor that each term is multiplied by to get the next term in the sequence.
The common ratio, often represented by the symbol \( r \), is vital in defining a geometric sequence.

• If \( r \) is positive, the sequence maintains its direction or increases, provided \( r > 1 \).
• If \( 0 < r < 1 \), the terms of the sequence decrease in magnitude.
• A negative \( r \) causes the sequence terms to alternate in sign.

Understanding the common ratio allows us to determine the nature and progression of the sequence.

Sequence Definition

A sequence is a set of numbers arranged in a specific order. Geometric sequences are a special type of sequence where each term after the first is the product of the previous term and a fixed, non-zero number, known as the common ratio.
Understanding this definition helps us identify geometric sequences, which have unique properties due to their multiplicative nature.

• For instance, the sequence \( 2, 6, 18, 54, \dots \) is geometric since each term is three times the previous one.
• This multiplicative pattern makes geometric sequences easy to spot once you check the ratios between terms.

Recognizing this can simplify analysis and processing of sequence problems.

Ratio Calculation

To determine if a sequence is geometric, examining the ratios between its consecutive terms is crucial. In a geometric sequence, these ratios must all be equal, making the term-by-term multiplication consistent.

Steps to Calculate Ratios:

• Take the second term and divide it by the first term to find the first ratio.
• Repeat this process for the subsequent terms.
• Compare these ratios to ascertain if they remain constant through the sequence.

For the sequence given in the exercise, the calculated ratios \( \frac{2}{3}, \frac{3}{4}, \frac{4}{5} \) varied, indicating that there is no constant common ratio, and therefore, the sequence is not geometric.
Understanding how to calculate and compare these ratios is key in identifying geometric sequences.