Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ a, a^{2}, a^{3}, a^{4}, \dots $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio \( r = a \).

Step 1: Identify the Sequence Type

The sequence provided is \( a, a^2, a^3, a^4, \ldots \). We need to check whether this sequence is geometric. A sequence is geometric if the ratio of any term to its previous term is constant.

Step 2: Calculate Ratios

Calculate the ratio of the second term to the first term: \( \frac{a^2}{a} = a \).Calculate the ratio of the third term to the second term: \( \frac{a^3}{a^2} = a \).Calculate the ratio of the fourth term to the third term: \( \frac{a^4}{a^3} = a \).

Step 3: Confirm Consistency

Since the ratio between consecutive terms is consistently \( a \), the sequence is geometric with a common ratio \( r = a \).

Step 4: Conclusion

The sequence \( a, a^2, a^3, a^4, \dots \) is geometric because each ratio of a term to the previous one is the same. The common ratio \( r \) is \( a \).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the key element that determines whether a sequence is geometric or not. To find the common ratio, you take any term in the sequence, divide it by its preceding term, and check if this quotient is the same for each pair of consecutive terms.

For instance, consider the sequence given: \( a, a^2, a^3, a^4, \ldots \).

• The ratio of the second term \( a^2 \) to the first term \( a \) is \( \frac{a^2}{a} = a \).
• The ratio of the third term \( a^3 \) to the second term \( a^2 \) is \( \frac{a^3}{a^2} = a \).
• Similarly, the ratio for the fourth term \( a^4 \) to the third term \( a^3 \) is \( \frac{a^4}{a^3} = a \).

This consistency in the ratio confirms it's a geometric sequence and the common ratio \( r \) is \( a \). Understanding and consistently applying this test is crucial for identifying geometric sequences.

Sequence Identification

Identifying the type of sequence you are dealing with is an essential step in solving any sequence-related problem. A sequence can generally be arithmetic, geometric, or neither.

• In arithmetic sequences, the difference between consecutive terms is constant.
• In geometric sequences, the ratio of consecutive terms is constant.
• Any sequence that does not fit these descriptions is neither.

In the exercise, after examining the sequence \( a, a^2, a^3, a^4, \ldots \), the consistent common ratio \( a \) indicates that the sequence is geometric. Successfully identifying the sequence type not only enables you to solve the problem at hand but also helps in predicting further terms in the sequence.

Algebra 2

In Algebra 2, understanding sequences, particularly geometric sequences, is integral to developing a broader comprehension of algebraic concepts.

• Algebra 2 students often learn about sequences in relation to exponential functions and series.
• They explore how the concept of rate changes fits into broader mathematical models.
• Engaging deeply with geometric sequences helps students appreciate the repeated multiplication structure that underlies many real-world phenomena.

In the sequence example \( a, a^2, a^3, a^4, \ldots \), Algebra 2 students learn not just to identify the common ratio but also to appreciate its application in different contexts. Recognizing patterns such as these forms a foundation for more advanced topics like calculus and mathematical modeling. Geometric sequences, therefore, are more than a mathematical exercise—they are a conceptual bridge to higher mathematics.