Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. \(0.8,4,20,100,500, \ldots\)

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of 5.

Step 1: Determine if the sequence is geometric

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant. Let's check the given sequence by dividing each term by the previous term to see if the ratio is constant.

Step 2: Calculate the ratios

Calculate the ratio of each successive term to check if it remains constant.\[ \text{Ratio}_1 = \frac{4}{0.8} = 5 \]\[ \text{Ratio}_2 = \frac{20}{4} = 5 \]\[ \text{Ratio}_3 = \frac{100}{20} = 5 \]\[ \text{Ratio}_4 = \frac{500}{100} = 5 \]

Step 3: Conclude whether the sequence is geometric

Since all the ratios are the same (5 in this case), the sequence is confirmed to be a geometric sequence.

Step 4: Identify the common ratio

The common ratio of the geometric sequence is the constant value we multiply by to get from one term to the next. Here, it is 5.

Key Concepts

Common Ratio

In the realm of geometric sequences, the common ratio is key. A geometric sequence is defined as one where each term is obtained by multiplying the previous term by a constant, known as the common ratio.
This remains consistent throughout the sequence.

• To determine the common ratio, divide any term in the sequence by its preceding term.
• If this division results in the same value for all consecutive terms, that value is the common ratio.

In our example sequence, calculating the ratios of each successive term, we find \(\frac{4}{0.8} = 5, \frac{20}{4} = 5, \frac{100}{20} = 5, \text{ and } \frac{500}{100} = 5\), consistently yields 5.
This confirms the sequence's nature as geometric with a common ratio of 5.

Sequence Analysis

Analyzing a sequence involves identifying patterns or principles that dictate the arrangement of numbers. In the context of a geometric sequence, we focus on the ratio between terms.
The process is systematic:

• Examine the differences in adjacent terms.
• Divide each term by the one before it, looking for constant results.
• Confirm consistency across multiple terms to establish a pattern.

In our exercise, the uniform ratios demonstrate the systematic nature of the sequence and provide evidence of its geometric properties.
Sequence analysis ensures we understand the foundational principles shaping the series.

Mathematical Sequences

Mathematical sequences are lists of numbers following a certain rule or pattern. They come in various types, each defined by the rules that generate them.

• Arithmetic Sequences: where each number is a certain sum away from the previous number.
• Geometric Sequences: where each number is a specific multiple of the previous one.
• Fibonacci Sequence: where each number is the sum of the two preceding ones.

Geometric sequences are particularly interesting because of their multiplicative nature, easily identified by their common ratio.
This characteristic allows for quick identification and verification as seen in the exercise.
Understanding these patterns not only helps in homework exercises but also in broader mathematical problem-solving.