- Calculate the difference between consecutive terms: \(-1 - 9 = -10\), \(-11 - (-1) = -10\), \(-21 - (-11) = -10\).
- Recognize that a constant difference of \(-10\) defines an arithmetic sequence.
- Confirm that the ratio is not constant, so it is not geometric.
- Final answer: The common difference is \(\boxed{-10}\).
Explanation
- Analyzing the sequence and finding the common difference
We have the sequence \(9, -1, -11, -21, \dots\). To identify the relationship between its terms, we first calculate the differences between consecutive terms:
- Difference 1: \(a_2 - a_1 = -1 - 9 = -10\)
- Difference 2: \(a_3 - a_2 = -11 - (-1) = -11 + 1 = -10\)
- Difference 3: \(a_4 - a_3 = -21 - (-11) = -21 + 11 = -10\)
Since the difference between consecutive terms is constant, we identify the sequence as an arithmetic sequence with a common difference of \(d = -10\).
- Verifying the sequence type
Next, we check for a potential common ratio to see if it is a geometric sequence. A geometric sequence would require \(r = \frac{a_2}{a_1} = \frac{a_3}{a_2}\).
- Ratio 1: \(\frac{a_2}{a_1} = \frac{-1}{9}\)
- Ratio 2: \(\frac{a_3}{a_2} = \frac{-11}{-1} = 11\)
Since the ratios are not equal, the sequence is not geometric. We have already confirmed it is an arithmetic sequence with a common difference of \(-10\). Thus, the correct description is that the common difference is \(-10\).
Examples
Arithmetic sequences are used everywhere in life, such as when calculating interest in a simple savings account, determining the cost of a long-distance taxi ride based on distance, or planning a schedule where tasks are spaced out by a fixed amount of time. If a taxi costs \$5 to start and \$2 per kilometer, your total cost follows an arithmetic sequence as you travel, allowing you to easily predict the fare for any distance. Identifying these patterns helps in budgeting, time management, and predicting future values in predictable, linear growth or decline.