To determine if a number is part of a quadratic sequence, you need to examine the sequence's differences. Quadratic sequences are characterized by a specific pattern in their differences: the first differences are not equal, but the second differences are constant. Here's a step-by-step breakdown:
Identifying a Quadratic Sequence
Understanding Differences
Quadratic sequences follow a specific pattern. The key lies in looking at the differences between terms.
- First Differences: Calculate the difference between consecutive terms in the sequence.
- For the sequence 1, 4, 9, 16, 25,... the first differences are 3, 5, 7, 9... These are not constant, so it's not a linear sequence.
- Second Differences: If the first differences are not equal, calculate the differences between the first differences. This is known as the second difference.
- For the sequence 1, 4, 9, 16, 25..., the second differences would be 2, 2, 2.... This is a constant difference.
- Key Point: According to the reference, quadratic sequences are identifiable by their constant second differences.
Example
Let's examine the sequence 2, 7, 14, 23, 34...
| Sequence | 2 | 7 | 14 | 23 | 34 |
|---|---|---|---|---|---|
| First Difference | 5 | 7 | 9 | 11 | |
| Second Difference | 2 | 2 | 2 |
In this example, the second difference is constant at 2. Therefore, the sequence 2, 7, 14, 23, 34... is a quadratic sequence.
Determining if a Number Belongs to the Sequence
Once you’ve identified a quadratic sequence, you need to know the rule for generating the sequence (the nth term formula), in order to establish whether any particular number belongs to it. This can be achieved using a system of equations based on the first few terms of the sequence.
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Find the nth Term: Use a general formula for quadratic sequence: an² + bn + c, where a, b, and c are constants.
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For the sequence 2, 7, 14, 23, 34...
- When n = 1: a(1)² + b(1) + c = 2
- When n = 2: a(2)² + b(2) + c = 7
- When n = 3: a(3)² + b(3) + c = 14
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Solve this system of equations to find the values of a, b, and c.
- a = 1
- b = 2
- c = -1
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Therefore, the nth term of the sequence is: n² + 2n - 1
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Test the Target Number: Once you have the nth term formula, if you substitute a positive integer into n, and you get the target number as an answer, then it is a number in the sequence. If you substitute all positive integers and never get the target number as the result, then it is not a number in the sequence.
- For our example, the sequence is n² + 2n - 1. Let's see if 35 is in the sequence.
- n² + 2n - 1 = 35 or n² + 2n - 36 = 0. This is a quadratic equation with non-integer answers for n, therefore, 35 is not in the sequence.
- Let's try 47. n² + 2n - 1 = 47 or n² + 2n - 48 = 0. Solving this will give us n = 6. This is a positive integer, so 47 is in the sequence, when n=6.
Key takeaways
- If the second differences are constant, it is a quadratic sequence.
- To confirm if a specific number is in the quadratic sequence, derive the sequence's rule (nth term) and test it.
