Which of the Following Is an Arithmetic Sequence?
In mathematics, an arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference.
To determine whether a sequence is an arithmetic sequence, we can check whether the difference between any two consecutive terms is the same. If the difference is the same for all pairs of consecutive terms, then the sequence is an arithmetic sequence.
Here are some examples of arithmetic sequences:
- 1, 3, 5, 7, 9
- 2, 6, 10, 14, 18
- -1, 0, 1, 2, 3
Here are some examples of sequences that are not arithmetic sequences:
- 1, 2, 4, 8, 16
- 3, 6, 9, 12, 15
- 1, 2, 4, 8, 16, 32
Questions
Here are some questions that can be asked about arithmetic sequences:
- What is the common difference of an arithmetic sequence?
- How can you find the nth term of an arithmetic sequence?
- How can you find the sum of an arithmetic series?
Discussion
The common difference of an arithmetic sequence is the difference between any two consecutive terms. It is a constant value that can be found by subtracting any two consecutive terms.
For example, the common difference of the sequence 1, 3, 5, 7, 9 is 2. This can be found by subtracting any two consecutive terms, such as 5 – 3 = 2 or 7 – 5 = 2.
The nth term of an arithmetic sequence can be found using the following formula:
a_n = a_1 + (n - 1)d where:
- a_n is the nth term
- a_1 is the first term
- d is the common difference
For example, the third term of the sequence 1, 3, 5, 7, 9 is 5. This can be found using the formula above, with a_1 = 1, d = 2, and n = 3.
a_3 = a_1 + (3 - 1)d a_3 = 1 + (2)2 a_3 = 5 The sum of an arithmetic series is the sum of all the terms in the sequence. It can be found using the following formula:
S_n = (n/2)(a_1 + a_n) where:
- S_n is the sum of the first n terms
- n is the number of terms
- a_1 is the first term
- a_n is the nth term
For example, the sum of the first 5 terms in the sequence 1, 3, 5, 7, 9 is 20. This can be found using the formula above, with n = 5.
S_5 = (5/2)(1 + 9) S_5 = (5/2)(10) S_5 = 25 Conclusion
Arithmetic sequences are a common type of sequence that is used in mathematics. They can be used to model a variety of real-world phenomena, such as the rise and fall of prices or the growth of populations.