Problem 65 - Algebra for College Students

Question

Which of the following sequences is not a geometric sequence? Explain your answer. a) \(1,2,4, \ldots\) b) \(0.1,0.01,0.001, \ldots\) c) \(-1,2,-4, \dots\) d) \(2,4,6, \dots\)

Step-by-Step Solution

Answer

Sequence (d) is not a geometric sequence because the ratio is not consistent.

Step 1 - Understanding Geometric Sequences

A geometric sequence is one where each term after the first is found by multiplying the previous term by a constant called the common ratio.

Step 2 - Check Sequence (a)

For sequence (a) \(1, 2, 4, \ldots\), find the common ratio by dividing the second term by the first term: \(\frac{2}{1} = 2\). Check the next term: \(\frac{4}{2} = 2\). The ratio is consistent, so it is a geometric sequence.

Step 3 - Check Sequence (b)

For sequence (b) \(0.1, 0.01, 0.001, \ldots\), find the common ratio: \(\frac{0.01}{0.1} = 0.1\). Check the next term: \(\frac{0.001}{0.01} = 0.1\). The ratio is consistent, so it is a geometric sequence.

Step 4 - Check Sequence (c)

For sequence (c) \(-1, 2, -4, \ldots\), find the common ratio: \(\frac{2}{-1} = -2\), then check the next term: \(\frac{-4}{2} = -2\). The ratio is consistent, so it is a geometric sequence.

Step 5 - Check Sequence (d)

For sequence (d) \(2, 4, 6, \ldots\), find the common ratio: \(\frac{4}{2} = 2\). Check the next term: \(\frac{6}{4} = 1.5\). The ratio is not consistent, so it is not a geometric sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element. It is the constant factor between consecutive terms. To find the common ratio, you take any term in the sequence and divide it by the previous term. For instance, let's examine sequence (a) from the exercise: \(1, 2, 4, \ldots\). If you divide the second term by the first term, \(\frac{2}{1} = 2\). Do the same for the next term, \(\frac{4}{2} = 2\). If the common ratio remains the same, then the sequence is geometric. This helps you identify and work with geometric sequences effectively.

Sequence Analysis

Sequence analysis involves examining the pattern of terms in a sequence to determine its nature. Consider sequence (d) from the exercise: \(2, 4, 6, \ldots\). To check if it is geometric, we find the common ratio by dividing consecutive terms: \(\frac{4}{2} = 2\). Then divide the next pair: \(\frac{6}{4} = 1.5\). Because the ratios differ (2 and 1.5), this tells us the sequence is not geometric. Analysis like this helps to differentiate between geometric sequences and other types such as arithmetic sequences, where the difference between terms is consistent, but the ratio is not.

Geometric Progression

A geometric progression is another term for a geometric sequence, where each term is derived by multiplying the previous term by the common ratio. Let's use sequence (c) from the exercise: \(-1, 2, -4, \ldots\). We notice the common ratio by dividing consecutive terms: \(\frac{2}{-1} = -2\) and \(\frac{-4}{2} = -2\). The consistent ratio of -2 confirms it is a geometric sequence. Understanding geometric progression helps solve various mathematical problems, especially in exponential growth contexts, financial calculations, and more.