Problem 13 - Algebra and Trigonometry

Question

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ 2,4,8,16, \dots $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of 2.

Step 1: Understand the Definition of a Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. We need to verify if this given sequence follows this pattern.

Step 2: Compare Consecutive Terms

Given the sequence: \(2, 4, 8, 16, \dots\), start by comparing the second term with the first term. Divide the second term by the first term: \[ \frac{4}{2} = 2 \]. This gives us a potential common ratio of 2.

Step 3: Verify the Common Ratio is Consistent

Check that the same ratio applies to the subsequent terms. Divide the third term by the second term: \(\frac{8}{4} = 2\), and the fourth term by the third term: \(\frac{16}{8} = 2\). Since all these calculations yield the same result, the common ratio is consistently 2.

Step 4: Conclusion

Since each consecutive pair of terms has a consistent ratio of 2, we can conclude that the sequence is indeed geometric with a common ratio of 2.

Key Concepts

Common Ratio

In the context of a geometric sequence, the common ratio is a fundamental element that dictates the entire progression of the sequence. It is the consistent factor by which we multiply each term to arrive at the next. For instance, if the common ratio is 2, as in the sequence 2, 4, 8, 16, each term is obtained by multiplying the previous term by 2.

To determine the common ratio, we perform a simple division of two consecutive terms. This ratio must remain the same throughout the sequence for it to be classified as geometric.

• Example: For the sequence 2, 4, 8, 16, we calculate \( \frac{4}{2} = 2 \), \( \frac{8}{4} = 2 \), and \( \frac{16}{8} = 2 \).
• If this calculation results in a consistent value across all consecutive terms, that consistent value is your common ratio.

Understanding and calculating the common ratio gives you the power to predict subsequent terms and verify the nature of the sequence.

Sequence Analysis

Sequence analysis involves examining the properties of a sequence to determine its structure and whether it follows certain rules, such as being arithmetic or geometric. Specifically, for a geometric sequence, analyzing involves checking the consistency of the common ratio.

By comparing consecutive terms, you identify whether multiplication by a constant (the common ratio) occurs. This analysis allows you to forecast future terms and understand relationships within the sequence. Here’s how you can perform a sequence analysis:

• Start by identifying the sequence you're analyzing. For example, 2, 4, 8, 16.
• Divide subsequent terms to check for a consistent common ratio.
• If consistent, confirm that the sequence is geometric.

This methodical analysis is useful for finding patterns and understanding how sequences are structured.

Mathematical Patterns

Recognizing mathematical patterns is a key skill in understanding sequences. A pattern in mathematics provides a visual or arithmetic depiction that can be traced to understand broader concepts. In the case of geometric sequences, the pattern is exponential growth, determined by the common ratio.

Consistency in such sequences is a powerful indicator. For example, the sequence 2, 4, 8, 16 showcases an easily recognizable pattern where each term doubles, indicating a consistent pattern dictated by the common ratio of 2.

• Identifying patterns helps to predict future numbers in the sequence.
• It aids in conceptualizing how growth or decay works within the sequence framework.

Understanding these patterns leads to a deeper comprehension of the underlying mathematical principles in various scenarios.