Finding the rule of an arithmetic sequence involves determining a formula that expresses any term in the sequence based on its position. This formula allows you to calculate any term without having to list all the preceding terms. The rule is usually expressed in one of two forms: explicit or recursive. Here, we'll focus on finding the explicit rule.
Understanding Arithmetic Sequences
An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'.
Finding the Explicit Rule
The explicit formula directly calculates any term in the sequence based on its position 'n'. The general form of the explicit formula is:
an = a1 + d(n - 1)
Where:
- an is the nth term of the sequence (the term you want to find).
- a1 is the first term of the sequence.
- d is the common difference between consecutive terms.
- n is the position of the term in the sequence (e.g., 1st, 2nd, 3rd, etc.).
Steps to Find the Explicit Rule:
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Identify the first term (a1): Look at the sequence and determine the value of the first term.
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Calculate the common difference (d): Subtract any term from its subsequent term. For example, d = a2 - a1, or d = a3 - a2. The difference should be the same between any two consecutive terms.
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Substitute a1 and d into the explicit formula: Once you have a1 and d, plug these values into the formula an = a1 + d(n - 1).
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Simplify the formula: Simplify the expression to obtain the explicit rule in its simplest form. This resulting formula represents the general rule for the sequence.
Example:
Consider the arithmetic sequence: 2, 5, 8, 11, ...
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First term (a1): a1 = 2
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Common difference (d): d = 5 - 2 = 3
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Substitute into the formula: an = 2 + 3(n - 1)
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Simplify: an = 2 + 3n - 3 = 3n - 1
Therefore, the explicit rule for this sequence is an = 3n - 1. You can use this formula to find any term in the sequence. For example, to find the 10th term (a10):
a10 = 3(10) - 1 = 30 - 1 = 29
Using the Rule
Once you have the rule, you can easily:
- Find any term in the sequence without having to list all the preceding terms.
- Predict future terms in the sequence.
- Analyze the behavior of the sequence as 'n' increases.
