To find the second difference of a quadratic sequence, you first need to calculate the first differences, and then calculate the differences between these first differences. Here's a step-by-step guide:
Understanding Quadratic Sequences
A quadratic sequence is a sequence where the n-th term can be expressed as an equation of the form an² + bn + c, where a, b, and c are constants, and a ≠ 0. The key characteristic of a quadratic sequence is that its second differences are constant, according to the provided reference which states, "In order to find the n th term (general term) of a quadratic sequence we have to find the second difference. To do this, we calculate the first difference between each term in a quadratic sequence and then calculate the difference between this new sequence."
Steps to Calculate the Second Difference
Here's how you find the second difference:
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Identify the sequence: Start with a quadratic sequence, for example: 2, 7, 14, 23, 34,...
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Calculate the first differences: Find the differences between consecutive terms:
- 7 - 2 = 5
- 14 - 7 = 7
- 23 - 14 = 9
- 34 - 23 = 11
The first difference sequence is: 5, 7, 9, 11,...
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Calculate the second differences: Now find the differences between the first differences:
- 7 - 5 = 2
- 9 - 7 = 2
- 11 - 9 = 2
The second difference sequence is: 2, 2, 2,...
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Observe the constant difference: If the second differences are the same, then the original sequence is confirmed to be a quadratic sequence and the second difference is that constant value. In this case, the second difference is 2.
Example Table
| Original Sequence | First Difference | Second Difference |
|---|---|---|
| 2 | ||
| 5 | ||
| 7 | 2 | |
| 7 | ||
| 14 | 2 | |
| 9 | ||
| 23 | 2 | |
| 11 | ||
| 34 | ||
| ... | ... | ... |
Practical Insights
- Constant Second Difference: A constant second difference is the key indicator of a quadratic sequence.
- Finding the nth Term: The second difference plays a vital role in determining the general formula (nth term) for a quadratic sequence.
- Verification: Calculating the second difference is a quick way to check if a sequence is quadratic or not.
By following these steps, you can easily find the second difference of any quadratic sequence.
