Problem 21 - 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. $$1.0,1.1,1.21,1.331, \ldots$$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of 1.1.

Step 1: Identify the Consecutive Terms

A geometric sequence is characterized by each term being obtained by multiplying the previous term by a constant called the common ratio, denoted as \( r \). Let's consider the given sequence: 1.0, 1.1, 1.21, 1.331. To determine whether this is a geometric sequence, we begin by identifying pairs of consecutive terms.

Step 2: Calculate Ratios Between Terms

Calculate the ratio between the first and second term: \( \frac{1.1}{1.0} = 1.1 \). Next, calculate the ratio between the second and third term: \( \frac{1.21}{1.1} \approx 1.1 \). Finally, calculate the ratio between the third and fourth term: \( \frac{1.331}{1.21} \approx 1.1 \).

Step 3: Verify Consistency of the Ratios

For the sequence to be geometric, the ratios obtained in Step 2 must be equal. All the calculated ratios (1.1, 1.1, 1.1) are indeed equal, which suggests that the sequence might be geometric.

Step 4: Confirm the Sequence is Geometric

Since all ratios between consecutive terms are consistently equal to 1.1, the sequence is confirmed to be geometric. The common ratio \( r \) is 1.1.

Key Concepts

Understanding the Common Ratio

The common ratio is a crucial part of understanding a geometric sequence. Think of it as a special number found between any two consecutive terms in the sequence.
This number remains constant throughout the series. This means every term is the result of multiplying the previous one by this ratio.
For instance, if you start with 1.0 and use a common ratio of 1.1, the next term is found by simply multiplying:

• 1.0 (first term) × 1.1 (common ratio) = 1.1 (second term)
• 1.1 (second term) × 1.1 (common ratio) = 1.21 (third term)

This pattern ensures the sequence unfolds in a predictable manner. Having a fixed common ratio helps us to quickly determine the progression of the sequence.

What Are Consecutive Terms?

Consecutive terms are just terms that follow one another in a sequence. In a typical set of numbers, these are the numbers that directly follow each other without gaps.
More simply, they are neighbors in the lineup of numbers, like links in a chain. Identifying consecutive terms is important in verifying if a sequence is geometric.
In the sequence provided (1.0, 1.1, 1.21, 1.331...), each pair of numbers sitting side by side is considered consecutive. Recognizing these helps us perform the essential ratio calculations to test the sequence's geometric nature.

Procedure for Sequence Verification

Sequence verification is the process where we ensure that an assumed property of a sequence, such as being geometric, is indeed true.
To verify if a sequence like 1.0, 1.1, 1.21, 1.331 is geometric, we attempt to make sure that the sense of consistency of having a constant common ratio holds across all the terms.

• First, we calculate the ratios between each pair of consecutive terms.
• Then, we compare these ratios.
• If all the calculated ratios are equal, and the pattern stays consistent, the sequence can be verified as geometric.

In our case, since all the ratios were 1.1, the sequence was successfully checked as geometric.

Performing Ratio Calculation

Ratio calculation is at the heart of determining whether a sequence is geometric.
To check the sequence 1.0, 1.1, 1.21, 1.331, start by dividing the second term by the first term:

• \( \frac{1.1}{1.0} = 1.1 \).
• Then, divide the third term by the second: \( \frac{1.21}{1.1} \approx 1.1 \).
• Finally, divide the fourth term by the third: \( \frac{1.331}{1.21} \approx 1.1 \).

All results consistently give us 1.1, confirming each consecutive pair shares the same ratio.
Ratios tell us how terms relate and progress, confirming fundamental characteristics like the common ratio in the pattern.