The rule an = an-1 + 8 can be used to find the next term of an arithmetic sequence. This rule states that to find any term in the sequence (an), you add 8 to the previous term (an-1). Let's explore this further.
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.
The Recursive Rule
The rule an = an-1 + 8 is an example of a recursive rule. A recursive rule defines each term based on the previous term.
- an: Represents the nth term in the sequence (the term you want to find).
- an-1: Represents the term immediately before an (the previous term).
- + 8: This is the constant difference, the common difference, which is 8 in this particular sequence.
Example: If the first term of the sequence is 2, then the second term is 2 + 8 = 10, the third term is 10 + 8 = 18, and so on.
The Explicit Rule
While the recursive rule is useful, it is also possible to define an arithmetic sequence using an explicit rule. The explicit rule expresses any term in the sequence directly in terms of its position (n). The general explicit formula of an arithmetic sequence is:
an = a1 + (n - 1)d
Where:
- an: The nth term of the sequence.
- a1: The first term of the sequence.
- n: The position of the term in the sequence.
- d: The common difference.
Applying it to our case: Assuming the first term of the sequence is ‘a1’ and the common difference ‘d’ is 8, we get an explicit rule as follows:
an = a1 + (n - 1)8
Comparing the Rules
| Rule Type | Formula | Description |
|---|---|---|
| Recursive | an = an-1 + 8 | Defines each term based on the previous term. Useful for finding the next term, given the previous one. |
| Explicit | an = a1 + (n - 1)8 | Defines each term directly by its position in the sequence. Useful for finding any specific term in the sequence. |
Practical Insights
- The recursive rule (an = an-1 + 8) is very useful for generating the sequence step by step, but it requires that you know the previous term.
- The explicit rule (an = a1 + (n - 1)8) is more efficient if you want to know a term in a specific position without computing all the preceding terms. For instance, to find the 100th term using the explicit rule, you can use the formula directly by substituting 100 for n.
