To find the sequence of an arithmetic sequence, you first need to confirm it is arithmetic, then determine the common difference, and finally, define the sequence using a formula.
Steps to Find the Arithmetic Sequence
Here's a breakdown of the process:
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Check if the Sequence is Arithmetic:
- An arithmetic sequence has a constant difference between consecutive terms.
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Calculate the Common Difference (d):
- The common difference (d) is the constant value added to each term to get the next term. According to the provided reference, the formula to calculate the common difference is:
d = a2 - a1 = a3 - a2 = ... = an - a(n-1) - Where:
- a2 is the second term in the sequence
- a1 is the first term in the sequence
- an is the nth term in the sequence
- a(n-1) is the term before the nth term.
- Example: In the sequence 2, 4, 6, 8, the common difference is 4-2 = 2.
- The common difference (d) is the constant value added to each term to get the next term. According to the provided reference, the formula to calculate the common difference is:
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Define the Sequence:
- You can define an arithmetic sequence using an explicit formula or a recursive formula.
- Explicit Formula (nth term):
an = a1 + (n - 1)d- Where:
- an is the nth term
- a1 is the first term
- n is the term number
- d is the common difference
- Where:
- Recursive Formula:
an = a(n-1) + d- Where:
- an is the nth term
- a(n-1) is the term before the nth term
- d is the common difference. You also need to know the first term, a1.
- Where:
Example
Let's say you have the sequence: 5, 8, 11, 14, ...
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Is it arithmetic? Yes, the difference between consecutive terms appears constant.
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Calculate the common difference (d):
- d = 8 - 5 = 3
- d = 11 - 8 = 3
- The common difference is 3.
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Define the sequence:
- Explicit Formula: an = 5 + (n - 1)3 which simplifies to an = 3n + 2
- Recursive Formula: an = a(n-1) + 3, a1 = 5
Therefore, you've successfully found the sequence by determining its common difference and expressing it as a formula.
