The nth term of a specific sequence, where n is greater than or equal to 2, can be found by subtracting the term that comes two positions before it from the immediately preceding term.
Here's a more detailed explanation:
Understanding the Sequence
The key characteristic of this sequence lies in its recursive nature. To generate any term (starting from the third term, n ≥ 2), we need to use the two terms that came before it.
How It Works
- Identify the Previous Terms: For any term (let's call it an), we need to know the term directly before it (an-1) and the term before that (an-2).
- Apply the Rule: The rule states that an = an-1 - an-2.
Example:
Let's assume the sequence starts with the terms 5 and 8. We can construct more terms using the subtraction rule:
| n | Term (an) | Calculation |
|---|---|---|
| 1 | 5 | Given |
| 2 | 8 | Given |
| 3 | a3 | 8 - 5 = 3 |
| 4 | a4 | 3 - 8 = -5 |
| 5 | a5 | -5 - 3 = -8 |
| 6 | a6 | -8 - (-5) = -3 |
| 7 | a7 | -3 - (-8) = 5 |
| 8 | a8 | 5 - (-3) = 8 |
| ... | ... | ... |
As you can see, the sequence would continue: 5, 8, 3, -5, -8, -3, 5, 8,... It repeats after 6 terms.
Practical Insights:
- Initial Terms are Important: The first two terms of the sequence define all the subsequent terms.
- Recursive Nature: The calculation of each term depends on the previous terms. This means to find term a10, we have to find a9 and a8, and so on.
- Pattern Recognition: These types of sequences may exhibit patterns or cycles, as shown in the above example.
Conclusion
The nth term of this particular sequence, for all n ≥ 2, is obtained by subtracting the (n-2)th term from the (n-1)th term: an = an-1 - an-2.
