Problem 51 - Prealgebra

Question

Describe the sequence using words. $$5,-10,20,-40,80, \dots$$

Step-by-Step Solution

Answer

The sequence is a geometric sequence where each term is multiplied by -2 to get the next term.

Step 1: Identify the Pattern

Examine the sequence: 5, -10, 20, -40, 80, ... to determine how each term relates to the previous term. Notice that each term after the first is a result of multiplying the previous term by a constant value.

Step 2: Determine the Common Ratio

Calculate the ratio between consecutive terms in the sequence. For example, the second term (-10) divided by the first term (5) is \\[-10 / 5 = -2\]. Similarly, the third term (20) divided by the second term (-10) equals \\[20 / -10 = -2\]. Continue this verification for other terms to confirm they all use the ratio of -2.

Step 3: Describe the Sequence Rule

With the common ratio of -2 confirmed, describe the sequence. It is a geometric sequence where each term is obtained by multiplying the previous term by -2. This alternating pattern where each term is double the previous term in absolute value but with alternating signs.

Key Concepts

Pattern Recognition

Pattern recognition is like being a detective for numbers.
When we look at the sequence 5, -10, 20, -40, 80, and so on, we need to find a consistent rule that applies to all the numbers. Identifying the pattern is the first step in understanding sequences.
In this sequence, we observe that each term changes in a regular way. This is an essential clue. When recognizing patterns, it's helpful to:

• Look for changes in signs, whether positive or negative.
• Check if numbers get larger or smaller systematically.
• Consider if each term is a multiplication of the previous term by a consistent value.

In this sequence, we quickly notice the signs alternate and the magnitude seems to double each time. Recognizing these patterns helps us analyze the series systematically and prepares us for finding a more detailed rule.

Common Ratio

The common ratio is the backbone of any geometric sequence. It tells us how much each term is multiplied by to get to the next term.
In our sequence: 5, -10, 20, -40, 80, ..., figuring out the common ratio is key.To find the common ratio:

• Pick two consecutive terms from the sequence.
• Divide the second term by the first one. This division gives you the common ratio.

Let's try it:- For the first two terms (5 and -10), we calculate: \(-10 / 5 = -2\).
- For the next pair, (-10 and 20), it is \(20 / -10 = -2\) again.Repeating this for other pairs verifies that the ratio is consistently -2. This common ratio of -2 tells us that each term is the previous term multiplied by -2. The negative sign indicates alternating directions between positive and negative.

Sequence Rule

Once we have recognized the pattern and determined the common ratio, we can establish the sequence rule.
The rule provides a formula or method to generate any term in the sequence without calculating all the previous terms. For this sequence, where we start with 5 and have confirmed a common ratio of -2, the sequence rule is: - The first term is 5. - Subsequent terms are found by multiplying the previous term by -2. This means, starting with the first term:

• Multiply 5 by -2 to get -10.
• Multiply -10 by -2 to get 20.
• Continue this process, multiplying each new term by -2.

This simple rule captures the essence of our sequence and shows how each term relates to the previous one, emphasizing both the magnitude and the alternating sign.