The nth term rule for a sequence of odd numbers is 2n - 1.
While the provided reference mentions that the nth term is n² + 1 for a different type of sequence, it also correctly lists the sequence of odd numbers as: 1, 3, 5, 7, 9... The formula to generate this sequence is not n² + 1. Let's understand how to derive the rule for odd numbers.
Understanding the Sequence of Odd Numbers
- The sequence of odd numbers starts with 1 and increases by 2 each time: 1, 3, 5, 7, 9, 11...
- This is an arithmetic sequence, where there is a common difference between successive terms. In this case, the common difference is 2.
Deriving the nth Term Rule
- Start with the Common Difference: The common difference is 2, so the nth term will involve multiplying n by 2 (2n).
- Adjust for the First Term: If we just use 2n, when n=1, we get 2. However, the first term is 1. To get the first term of the sequence (1), we need to subtract 1 from 2n, thus making it 2n - 1.
Example
| n | 2n - 1 | Term |
|---|---|---|
| 1 | 2(1) - 1 | 1 |
| 2 | 2(2) - 1 | 3 |
| 3 | 2(3) - 1 | 5 |
| 4 | 2(4) - 1 | 7 |
| 5 | 2(5) - 1 | 9 |
This confirms that the correct nth term rule for the sequence of odd numbers is indeed 2n - 1.
