Question

In Exercises 5–12, tell whether the sequence is geometric. Explain your reasoning. \(5,20,35,50,65, \ldots\)

Short answer

No, the sequence \(5, 20, 35, 50, 65, \ldots\) is not a geometric sequence because there is no common ratio between sequential terms. Instead, the sequence is an arithmetic sequence as there is a fixed constant difference of 15 between sequential terms.

Step-by-step solution

Identify the common difference

To begin with, identify the common difference between sequential terms in the sequence. That is, subtract the second term from the first, the third term from the second, and so forth. For the given sequence \(5, 20, 35, 50, 65, \ldots\), this would be: 20 - 5 = 15, 35 - 20 = 15, 50 - 35 = 15, and so on.

Check if there is a common ratio

After identifying the common difference, check if there is a common factor that the preceding term is multiplied by to obtain the next term. Divide the second term by the first term, the third term by the second term, and so forth. For the sequence given, this would be: 20 / 5 = 4, 35 / 20 = 1.75, 50 / 35 = 1.43, etc.

Conclusion

Since the ratios between the terms are not constant, it implies that the sequence \(5, 20, 35, 50, 65, \ldots\) is not a geometric sequence. It is, however, an arithmetic sequence as there is a fixed constant difference between sequential terms (15 in this case).

Key concepts

Arithmetic Sequences

Understanding arithmetic sequences is essential in recognizing patterns in various numerical problems. An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant value, known as the common difference, to the previous term.

For instance, consider the sequence 3, 7, 11, 15, which is arithmetic because each term is 4 more than the one before it. In general, an arithmetic sequence can be written as: $$\begin{align*} a, \ a + d, \ a + 2d, \ a + 3d, \ \text{...} \ a + (n-1)d, \ \text{where} \ a & = \text{the first term,} \ d & = \text{common difference,} \ n & = \text{the number of terms.} \ \text{...} \ \text{where} \ a & = \text{the first term,} \ d & = \text{common difference,} \ n & = \text{the number of terms.} \ \text{...} \ \text{where} \ a & = \text{the first term,} \ d & = \text{common difference,} \ n & = \text{the number of terms.} \ \text{...} \$$ The ability to identify an arithmetic sequence enables a student to forecast future terms or even find missing values within the sequence.

Common Difference

The common difference is a key feature of arithmetic sequences. It's the unvarying difference between any two consecutive terms. When analyzing sequences, identifying the common difference is a primary step toward understanding their nature.

In our example of the sequence 5, 20, 35, 50, 65, ..., the common difference is obtained by subtracting one term from the next, consistently resulting in 15. This constant value allows for a quick assessment that the sequence is arithmetic. The common difference doesn't just offer a way to recognize the sequence, but also provides a formulaic approach to finding any term within the sequence.

Common Ratio

In contrast to arithmetic sequences, geometric sequences rely on a common ratio rather than a common difference. Each term in a geometric sequence is found by multiplying the previous term by a constant called the common ratio.

To determine if a sequence is geometric, you look at the ratios of successive terms (divide one term by the one before it). If this ratio is the same for all pairs of consecutive terms, then you have a geometric sequence. For example, the sequence 2, 6, 18, 54, ... is geometric with a common ratio of 3, since 6/2 = 18/6 = 54/18 = 3. Understanding the concept of the common ratio is essential for solving problems related to geometric growth and for sequence analysis in general.

Sequence Analysis

Sequence analysis involves examining the relationship between elements in a series to determine its nature and predict subsequent terms. It's a cornerstone in understanding mathematical patterns and progressions.

Arithmetic and geometric sequences are foundational concepts, and analyzing them requires different approaches. Checking for a common difference helps identify arithmetic sequences, whereas finding a consistent common ratio flags geometric sequences. Sequence analysis isn't confined to simply determining the type of sequence; it also includes finding specific terms, the sum of terms, and applying the information to real-world scenarios. Effective analysis relies on keen observation and application of these principles to draw meaningful conclusions about the sequence at hand.