Finding the nth term of a sequence is a key concept in algebra. It allows you to predict any term in a sequence without having to list all the terms before it. For arithmetic sequences (sequences with a constant difference between consecutive terms), the process is straightforward.
Understanding Arithmetic Sequences
An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (d).
- Example: In the sequence 2, 5, 8, 11..., the common difference is 3 (5-2 = 3, 8-5 = 3, and so on).
Finding the nth Term Formula
The formula for the nth term (an) of an arithmetic sequence is:
an = a1 + (n - 1)d
Where:
- an is the nth term you want to find.
- a1 is the first term of the sequence.
- n is the position of the term you want to find (e.g., 1st, 2nd, 3rd, ...).
- d is the common difference.
Step-by-Step Process
- Identify the common difference (d): Subtract any term from the term that follows it.
- Identify the first term (a1): This is simply the first number in the sequence.
- Substitute into the formula: Plug the values of a1 and d into the formula an = a1 + (n - 1)d.
- Simplify: The resulting expression will be the formula for the nth term.
Example
Let's find the nth term of the sequence 2, 5, 8, 11...
- Common difference (d): 5 - 2 = 3
- First term (a1): 2
- Substitute: an = 2 + (n - 1)3
- Simplify: an = 2 + 3n - 3 = 3n - 1
Therefore, the nth term of this sequence is 3n - 1. To find the 10th term, you would substitute n = 10: a10 = 3(10) - 1 = 29.
This process is applicable to most arithmetic sequences encountered at the Grade 8 level. More complex sequences might require different approaches, but the fundamental principle remains the same: find a pattern and express it as a formula.
