Problem 82 - College Algebra with Modeling and Visualization

Question

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$0.7,0.21,0.063,0.0189,0.00567$$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of 0.3.

Step 1: Identify Consistent Pattern

To determine the sequence type, first check if there is a consistent difference or ratio between consecutive terms. Let's find the ratio by dividing each term by the previous term.

Step 2: Calculate Ratio Between Terms

Calculate the ratio between the first and second term: \[\frac{0.21}{0.7} = 0.3\]Now, calculate the ratio between the second and third term: \[\frac{0.063}{0.21} = 0.3\]Continue to calculate for the next terms: \[\frac{0.0189}{0.063} = 0.3\]and \[\frac{0.00567}{0.0189} = 0.3\]

Step 3: Determine Type of Sequence

Since the ratio between each consecutive term is constant, the sequence is geometric. The common ratio is 0.3.

Key Concepts

Geometric Sequence

In the wonderful world of sequences, the geometric sequence holds a special place. Imagine a sequence where each term progressively transforms into the next by multiplying with a consistent factor. That's a geometric sequence for you! It's like a chain of events where every next step is predictable and derived by multiplying the previous term by a fixed number, known as the common ratio.
A geometric sequence can start with any number, and it progresses by this multiplication, allowing for either steady growth if the common ratio is greater than one, or a diminishing value if it lies between zero and one. Each term is a result of the previous term being multiplied by the common ratio, so understanding this consistent pattern is key to unraveling future terms or even past ones in the sequence.

Common Ratio

The common ratio is the cornerstone of a geometric sequence. It is the constant factor you multiply with to get from one term to the next. To find the common ratio, simply divide any term by its preceding term.

• If the ratio remains consistent across all terms, you indeed have a geometric sequence.
• When the common ratio is positive, the sequence will grow or shrink steadily.
• If the common ratio is negative, you will observe an alternating pattern between the terms, flipping signs as you go along the sequence.

In our given sequence, we calculated the common ratio by dividing consecutive terms, and discovered it to be 0.3. This tells us that each term in the sequence is formed by multiplying the previous term by 0.3, confirming the geometric nature of the sequence.

Finite Sequence

Not all sequences extend forever; some have a designated ending, and these are known as finite sequences. Finite sequences have a specific number of terms. This means if you lay them out one by one, there is a last term you can identify. Finite sequences are often easier to analyze than infinite sequences because you have access to all terms.
In the case of our exercise, the sequence is finite, consisting of five precise terms: 0.7, 0.21, 0.063, 0.0189, and 0.00567. When dealing with finite sequences, you know exactly where they stop, which can make calculations more straightforward and allows for clear understanding of the sequence's structure and value changes across its span.