The sequence of numbers provided, 35.8, 43.2, 50.6, 58, appears to be a series of values that are increasing in a somewhat linear fashion, but with a slight irregularity. To understand the nature of this sequence, let's examine the differences between each pair of consecutive numbers.
Analysis of the Sequence
Calculating the differences between each pair of consecutive numbers can give us insight into the pattern or rule governing the sequence. The differences are as follows: - Between 35.8 and 43.2, the difference is 7.4. - Between 43.2 and 50.6, the difference is 7.4. - Between 50.6 and 58, the difference is 7.4. Given that the differences between each pair of consecutive numbers are constant (7.4), this indicates that the sequence is arithmetic.
Characteristics of an Arithmetic Sequence
An arithmetic sequence is a sequence of numbers such that the difference between any two successive members is constant. This constant difference is called the common difference. In the case of the given sequence, the common difference is 7.4. The formula for the nth term of an arithmetic sequence is given by: a_n = a_1 + (n-1)d, where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.
| Term Number | Term Value |
|---|---|
| 1 | 35.8 |
| 2 | 43.2 |
| 3 | 50.6 |
| 4 | 58 |
To predict the next term in the sequence, we would add the common difference (7.4) to the last known term (58). Thus, the next term would be 58 + 7.4 = 65.4. This method can be used to generate as many terms of the sequence as needed.
Applications of Arithmetic Sequences
Arithmetic sequences have a wide range of applications. They can be used to model real-world phenomena where a constant change occurs over a period, such as population growth, financial transactions, and physical movements. For instance, if a car is traveling at a constant acceleration, the distance it covers over equal intervals of time forms an arithmetic sequence.
Real-World Example
Consider a company that increases its production output by 100 units every month. If the company starts with a production output of 500 units in the first month, the sequence of production outputs for the subsequent months would be an arithmetic sequence with a first term of 500 and a common difference of 100. Thus, the production outputs for the first four months would be 500, 600, 700, and 800 units, respectively.
- First month: 500 units
- Second month: 500 + 100 = 600 units
- Third month: 600 + 100 = 700 units
- Fourth month: 700 + 100 = 800 units
What is the formula for the nth term of an arithmetic sequence?
+The formula for the nth term of an arithmetic sequence is a_n = a_1 + (n-1)d, where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.
How do you find the next term in an arithmetic sequence?
+To find the next term in an arithmetic sequence, you add the common difference to the last known term. For example, if the last term is 58 and the common difference is 7.4, the next term would be 58 + 7.4 = 65.4.
In conclusion, the sequence 35.8, 43.2, 50.6, 58 is an arithmetic sequence with a common difference of 7.4. Understanding arithmetic sequences and their applications can provide valuable insights into patterns and changes in various fields, from mathematics and physics to economics and engineering.
