How to Calculate Quadratic nth Term?
Finding the nth term of a quadratic sequence involves a systematic approach using differences. A quadratic sequence follows a pattern where the second differences are constant. The general form of a quadratic nth term is expressed as: an² + bn + c, where 'a', 'b', and 'c' are constants we need to determine.
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Calculate the first differences: Subtract each term from the next term in the sequence.
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Calculate the second differences: Subtract each first difference from the next first difference. These second differences should be constant for a true quadratic sequence. This constant value is 2a.
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Solve for 'a': Divide the constant second difference by 2:
a = (constant second difference) / 2 -
Find a simplified sequence: Subtract the
an²sequence from the original quadratic sequence. This will result in a linear sequence. -
Find the nth term of the simplified linear sequence: This can be done using the formula for the nth term of an arithmetic progression:
dn + kwhere 'd' is the common difference and 'k' is a constant. -
Determine 'b' and 'c': The nth term of the linear sequence represents
bn + c. Therefore, 'b' is the common difference and 'c' is the constant term. -
Assemble the nth term: Substitute the values of 'a', 'b', and 'c' into the general formula: an² + bn + c.
Example
Let's consider the quadratic sequence: 3, 8, 15, 24, 35...
- First differences: 5, 7, 9, 11...
- Second differences: 2, 2, 2... (Constant, confirming it's quadratic)
- Solve for 'a':
a = 2 / 2 = 1 - Simplified sequence (subtracting n²): 3-1, 8-4, 15-9, 24-16, 35-25... which simplifies to 2, 4, 6, 8, 10...
- Nth term of the simplified linear sequence: The nth term is
2n(common difference = 2, constant term = 0) - Determine 'b' and 'c': 'b' = 2, 'c' = 0
- Assemble the nth term: The nth term of the original quadratic sequence is
1n² + 2n + 0or simplyn² + 2n.
