Which Formula Can Be Used To Describe The Sequence
In mathematics, a sequence is an ordered list of numbers. A geometric sequence is a sequence in which each term is equal to the previous term multiplied by a constant ratio. The constant ratio is called the common ratio.
The general formula for a geometric sequence is:
a_n = a_1 r^{n - 1} where:
- $a_n$ is the $n$th term of the sequence
- $a_1$ is the first term of the sequence
- $r$ is the common ratio
For example, the sequence 1, 2, 4, 8, 16 is a geometric sequence with first term 1 and common ratio 2.
Questions Related to Which Formula Can Be Used To Describe The Sequence
Here are some questions that can be asked about which formula can be used to describe a sequence:
- What is the difference between an arithmetic sequence and a geometric sequence?
- How can you determine whether a given sequence is arithmetic or geometric?
- How can you find the common ratio of a geometric sequence?
- How can you use the formula for a geometric sequence to find the value of a specific term in the sequence?
Discussion of Questions
Arithmetic vs. Geometric Sequences
The main difference between an arithmetic sequence and a geometric sequence is that the difference between consecutive terms in an arithmetic sequence is constant, while the ratio of consecutive terms in a geometric sequence is constant.
In an arithmetic sequence, each term is equal to the previous term plus a constant difference. The constant difference is called the common difference.
The general formula for an arithmetic sequence is:
a_n = a_1 + d(n - 1) where:
- $a_n$ is the $n$th term of the sequence
- $a_1$ is the first term of the sequence
- $d$ is the common difference
For example, the sequence 1, 3, 5, 7, 9 is an arithmetic sequence with first term 1 and common difference 2.
In a geometric sequence, each term is equal to the previous term multiplied by a constant ratio. The constant ratio is called the common ratio.
The general formula for a geometric sequence is:
a_n = a_1 r^{n - 1} where:
- $a_n$ is the $n$th term of the sequence
- $a_1$ is the first term of the sequence
- $r$ is the common ratio
For example, the sequence 1, 2, 4, 8, 16 is a geometric sequence with first term 1 and common ratio 2.
Determining Whether a Sequence Is Arithmetic or Geometric
To determine whether a given sequence is arithmetic or geometric, you can look for a pattern in the difference between consecutive terms. If the difference between consecutive terms is constant, then the sequence is arithmetic. If the ratio of consecutive terms is constant, then the sequence is geometric.
For example, the sequence 1, 3, 5, 7, 9 is arithmetic because the difference between consecutive terms is constant at 2. The sequence 1, 2, 4, 8, 16 is geometric because the ratio of consecutive terms is constant at 2.
Finding the Common Ratio of a Geometric Sequence
The common ratio of a geometric sequence can be found by dividing any term in the sequence by the term before it.
For example, in the sequence 1, 2, 4, 8, 16, the common ratio is 2 because 2 / 1 = 2, 4 / 2 = 2, 8 / 4 = 2, and 16 / 8 = 2.
Finding the Value of a Specific Term in a Geometric Sequence
The value of a specific term in a geometric sequence can be found using the formula for a geometric sequence.
For example, to find the value of the 10th term in the sequence 1, 2, 4, 8, 16, you would use the following formula:
a_{10} = 1 \cdot 2^{10 - 1} = 1024 Therefore, the value of the 10th term in the sequence is 1024.