Problem 85 - College Algebra

Question

Exercises \(85-87\) will help you prepare for the material covered in the next section. Consider the sequence \(1,-2,4,-8,16, \ldots .\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}}\)and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

Step-by-Step Solution

Answer

The ratio between each term and its predecessor in the sequence is a constant factor of -2.

Step 1: Identify the sequence

The sequence provided is \(1,-2,4,-8,16, \ldots \) This is an alternating sequence, with each term doubling in magnitude and changing sign from the previous term.

Step 2: Calculate Ratios

Calculate the ratio of consecutive elements of the sequence. This will give \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}} \) and \(\frac{a_{5}}{a_{4}}\). Doing this calculation results in \(-2, -2, -2, -2\).

Step 3: Observe the pattern

The pattern in the ratios is a constant factor of -2 which means that each term in the sequence is obtained by multiplying the previous term by -2.

Key Concepts

Alternating Sequence

An alternating sequence is a type of sequence where the signs of the terms alternate from positive to negative or vice versa. In our example with the sequence \(1, -2, 4, -8, 16, \ldots \), each term changes its sign compared to the previous one.
This happens because each term in the sequence is the opposite sign of the term before it. This alternating pattern is easy to spot and is one of the key characteristics. Alternating sequences can create unique visual patterns and are often used in mathematical models to represent oscillating behavior.

Ratio of Consecutive Terms

To understand the relationship between terms in a sequence, we often calculate the ratio of consecutive terms. This means we divide one term by the term that comes before it.
For our sequence \(1, -2, 4, -8, 16, \ldots \), we calculate each pair's ratio as follows:

• \(\frac{a_{2}}{a_{1}} = \frac{-2}{1} = -2\)
• \(\frac{a_{3}}{a_{2}} = \frac{4}{-2} = -2\)
• \(\frac{a_{4}}{a_{3}} = \frac{-8}{4} = -2\)
• \(\frac{a_{5}}{a_{4}} = \frac{16}{-8} = -2\)

Each of these ratios is \(-2\), indicating a consistent relationship between the terms. This observation helps highlight the regular transformation that occurs in the sequence.

Constant Ratio

When the ratio of consecutive terms in a sequence remains constant, the sequence is a geometric sequence. This is the case with our sequence, where we found that each ratio is \(-2\).
A constant ratio tells us that each term is obtained by multiplying the previous term by the same factor. For this sequence, every term is obtained by multiplying the prior term by \(-2\). This constant ratio simplifies the exploration of such sequences because it gives a predictable pattern for their progression. Identifying this constant factor is crucial for understanding and predicting the behavior of geometric sequences.

Sequence Pattern

The overall pattern of the sequence provides insight into the nature and behavior of the sequence. In \(1, -2, 4, -8, 16, \ldots\), the sequence pattern is dictated by the alternating signs and a constant multiplication factor of \(-2\).
Understanding the pattern allows us to predict future terms and comprehend the nature of the sequence. Here, we know that:

• The sequence alternates in sign from positive to negative and vice versa.
• The sequence grows or decreases by a factor of 2 in absolute value.
• Each term is a result of multiplying the previous term by \(-2\).

Recognizing these patterns is critical in solving problems involving sequences, allowing us to generalize their behaviors and make predictions about further terms.