An example of an arithmetic sequence's general term can be derived from the sequence {4, 8, 12, 16...}.
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.
- Common Difference: In the given example {4, 8, 12, 16...}, the common difference is 4, as you add 4 to each term to get the next one.
Deriving the General Term
To determine the general term of an arithmetic sequence, we use the formula:
an = a1 + (n - 1)d
Where:
- an is the nth term of the sequence
- a1 is the first term of the sequence
- n is the position of the term in the sequence
- d is the common difference
Let's apply this to the example: {4, 8, 12, 16...}
- a1 (first term) = 4
- d (common difference) = 4
So the general term is:
an = 4 + (n - 1)4
Simplifying:
an = 4 + 4n - 4
an = 4n
Therefore, a general term for this sequence would be 4n.
Verifying with Examples
To verify the general term is correct:
| n (term number) | Calculation using 4n | Result |
|---|---|---|
| 1 | 4 * 1 | 4 |
| 2 | 4 * 2 | 8 |
| 3 | 4 * 3 | 12 |
| 4 | 4 * 4 | 16 |
This matches the given sequence {4, 8, 12, 16...}, confirming that 4n is the general term for this specific arithmetic sequence.
