A quadratic sequence can be justified by demonstrating that the second difference between consecutive terms is constant. This constant second difference is a key characteristic that distinguishes quadratic sequences from arithmetic or other types of sequences.
Here's a breakdown of how to justify a quadratic sequence:
1. Understanding Sequences and Differences
Before justifying a quadratic sequence, it's important to understand the concept of differences in a sequence.
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Sequence: A sequence is a list of numbers arranged in a specific order (e.g., 2, 5, 10, 17, 26...).
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First Difference: The first difference is found by subtracting each term from the term that follows it.
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Second Difference: The second difference is found by calculating the difference between consecutive first differences.
2. The Constant Second Difference
The defining characteristic of a quadratic sequence is its constant second difference. If the second difference between consecutive terms is the same, then the sequence is quadratic.
3. Justification Process: An Example
Let's consider the sequence: 3, 7, 13, 21, 31...
Step 1: Calculate the First Differences
Subtract each term from the next:
- 7 - 3 = 4
- 13 - 7 = 6
- 21 - 13 = 8
- 31 - 21 = 10
So, the first differences are: 4, 6, 8, 10...
Step 2: Calculate the Second Differences
Subtract each first difference from the next:
- 6 - 4 = 2
- 8 - 6 = 2
- 10 - 8 = 2
So, the second differences are: 2, 2, 2...
Step 3: Justification
Since the second differences are constant (all equal to 2), we can justify that the sequence 3, 7, 13, 21, 31... is a quadratic sequence.
4. General Form of a Quadratic Sequence
A quadratic sequence can generally be represented by the formula:
- an2 + bn + c
where a, b, and c are constants, and n represents the term number (e.g., 1, 2, 3...). Once you've established the sequence is quadratic via constant second differences, you could attempt to derive the specific formula, although this isn't strictly necessary to "justify" it.
5. Summary
To justify a sequence as quadratic, you must show that the second difference between its terms is constant. Calculate the first differences and then the second differences. If the second differences are all equal, the sequence is quadratic.
