Question
What kind of sequence is this? 86, –172, 344, –688, ...
Answers
Answered by
GPT-4o mini
To determine the type of sequence represented by the numbers \(86, -172, 344, -688, \ldots\), let's analyze the pattern:
1. **Examine the first few terms:**
- \(a_1 = 86\)
- \(a_2 = -172\)
- \(a_3 = 344\)
- \(a_4 = -688\)
2. **Calculate the ratio between consecutive terms:**
- \( \frac{a_2}{a_1} = \frac{-172}{86} = -2 \)
- \( \frac{a_3}{a_2} = \frac{344}{-172} = -2 \)
- \( \frac{a_4}{a_3} = \frac{-688}{344} = -2 \)
The ratio between consecutive terms is consistently \(-2\). This indicates that each term is obtained by multiplying the previous term by \(-2\).
3. **Identify the type of sequence:**
- Since each term is generated from the previous one by multiplying by a fixed ratio (in this case, \(-2\)), this sequence is a **geometric sequence**. The common ratio is \(-2\).
Thus, the sequence \(86, -172, 344, -688, \ldots\) is a geometric sequence with a common ratio of \(-2\).
1. **Examine the first few terms:**
- \(a_1 = 86\)
- \(a_2 = -172\)
- \(a_3 = 344\)
- \(a_4 = -688\)
2. **Calculate the ratio between consecutive terms:**
- \( \frac{a_2}{a_1} = \frac{-172}{86} = -2 \)
- \( \frac{a_3}{a_2} = \frac{344}{-172} = -2 \)
- \( \frac{a_4}{a_3} = \frac{-688}{344} = -2 \)
The ratio between consecutive terms is consistently \(-2\). This indicates that each term is obtained by multiplying the previous term by \(-2\).
3. **Identify the type of sequence:**
- Since each term is generated from the previous one by multiplying by a fixed ratio (in this case, \(-2\)), this sequence is a **geometric sequence**. The common ratio is \(-2\).
Thus, the sequence \(86, -172, 344, -688, \ldots\) is a geometric sequence with a common ratio of \(-2\).