Arkell decorates old bottles to make vases. The table shows the number of vases he has.
| Day | Vases |
|---|---|
| 1 | 6 |
| 2 | 12 |
| 3 | 18 |
| 4 | 24 |
| 5 | 30 |
This situation represents a ______ sequence. The common ______ is ______. At the end of the seventh day, Arkell will have ______ vases.
Answer (2)
The sequence of vases is arithmetic because the difference between consecutive terms is constant.
The common difference is 6.
The formula for the nth term of an arithmetic sequence is a n = a 1 + ( n − 1 ) d .
At the end of the seventh day, Arkell will have 42 vases.
Explanation
Analyzing the Data First, let's analyze the given data. The number of vases Arkell makes each day is: Day 1: 6, Day 2: 12, Day 3: 18, Day 4: 24, Day 5: 30. We need to determine if this is an arithmetic or geometric sequence, find the common difference or ratio, and then find the number of vases at the end of the seventh day.
Checking for Arithmetic Sequence To determine if the sequence is arithmetic, we check if the difference between consecutive terms is constant. The differences are: 12 - 6 = 6, 18 - 12 = 6, 24 - 18 = 6, 30 - 24 = 6. Since the difference is constant, the sequence is arithmetic.
Finding the Common Difference The common difference, d , is 6.
Using the Arithmetic Sequence Formula Now, we need to find the number of vases at the end of the seventh day. We use the formula for the nth term of an arithmetic sequence: a n = a 1 + ( n − 1 ) d , where a 1 is the first term, n is the term number, and d is the common difference. In this case, a 1 = 6 , n = 7 , and d = 6 .
Calculating the 7th Term Plugging in the values, we get: a 7 = 6 + ( 7 − 1 ) × 6 = 6 + 6 × 6 = 6 + 36 = 42 .
Final Answer Therefore, at the end of the seventh day, Arkell will have 42 vases.
Examples
Understanding arithmetic sequences is useful in many real-life situations. For example, if you save a fixed amount of money each month, the total amount you have saved over time forms an arithmetic sequence. Similarly, if a taxi charges a fixed initial fee plus a fixed amount per mile, the total cost of the ride forms an arithmetic sequence. Recognizing these patterns can help you predict future values and make informed decisions.
The sequence of vases is an arithmetic sequence with a common difference of 6. By using the formula for the nth term of an arithmetic sequence, we find that at the end of the seventh day, Arkell will have 42 vases. Therefore, the completed statements are: The situation represents an arithmetic sequence, the common difference is 6, and at the end of the seventh day, Arkell will have 42 vases.
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