Problem 12 - College Algebra

Question

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. $$ 6,8,11,15,20, \ldots $$

Step-by-Step Solution

Answer

The sequence is not geometric; it lacks a constant common ratio.

Step 1: Understand the Definition of a Geometric Sequence

A sequence is called geometric if each term after the first is the result of multiplying the previous term by a fixed, non-zero number called the common ratio.

Step 2: Check for a Common Ratio

To determine whether the sequence is geometric, we need to check if there is a constant ratio between consecutive terms.1. Calculate the ratio between the second and the first term: \( \frac{8}{6} = \frac{4}{3} \).2. Calculate the ratio between the third and the second term: \( \frac{11}{8} \approx 1.375 \).3. Calculate the ratio between the fourth and the third term: \( \frac{15}{11} \approx 1.364 \).4. Calculate the ratio between the fifth and the fourth term: \( \frac{20}{15} = \frac{4}{3} \).Since the ratio is not constant, the sequence is not geometric.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key feature. It refers to the constant factor by which each term in the sequence is multiplied to get the next term. When you want to identify or verify if a sequence is geometric, the common ratio is crucial. For instance, in a geometric sequence where the first term is 2 and the common ratio is 3, the sequence progresses as:

• First term: 2
• Second term: 2 multiplied by 3 equals 6
• Third term: 6 multiplied by 3 equals 18
• And so on: 54, 162...

In any true geometric sequence, this ratio remains consistent from one term to the next. When you calculate the common ratio for each term in the sequence from the problem:

• Ratio of second to first term: \( \frac{8}{6} \)
• Ratio of third to second term: \( \frac{11}{8} \)
• And so forth.

The non-constant results show us that a consistent common ratio doesn’t exist here, confirming the sequence is not geometric.

Sequence Analysis

Sequence analysis involves examining the properties and patterns of sequences. It's essential for identifying whether a sequence fits a particular type, like geometric or arithmetic. In the exercise given, the method involved checking if a fixed common multiplier, or ratio, could generate the sequence:

• Identify the first few terms: 6, 8, 11, 15, 20...
• Calculate successive term ratios: \( \frac{8}{6}, \frac{11}{8}, \frac{15}{11}, \frac{20}{15} \).
• Observe inconsistencies in these ratios.

When undertaking sequence analysis, it is also critical to consider other sequence types in case it does not fit the geometric model. Since the ratios in this case vary, it confirms the sequence analysis outcome that it is not a geometric sequence.

Mathematical Proof

Mathematical proof is a logical argument verifying the truth of a mathematical statement. To prove whether a sequence is geometric, one can employ a series of calculations involving the common ratio. Here’s a step-by-step approach to how proof was conducted in this exercise:

• Start by calculating the ratio between the first and second term.
• Continue calculating ratios for each subsequent pair of terms.
• Compare these ratios.
• If they are identical, it proves the sequence is geometric.
• If not, as in this case, the sequence is not geometric, proven through the inconsistent ratios.

Using this method provides a clear mathematical proof that is crucial for validating the type of sequence. This proof helps to establish facts based on logical reasoning and defined mathematical procedures, underscoring its importance in understanding sequences.