A polynomial sequence whose terms have constant second differences is a quadratic polynomial. This can be represented in the general form:
\[ P(n) = an^2 + bn + c \]
where \(a\), \(b\), and \(c\) are constants, and \(n\) represents the term number (e.g., \(n = 0, 1, 2, \ldots\)).
To demonstrate that the second differences of a quadratic polynomial are constant, consider the steps below:
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Calculate the first differences of the sequence generated by the polynomial: \[ \Delta P(n) = P(n+1) - P(n) \]
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Calculate the second differences, which are the differences of the first differences: \[ \Delta^2 P(n) = \Delta P(n+1) - \Delta P(n) \]
Let's illustrate this with a specific quadratic polynomial. Take \( P(n) = n^2 \):
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Calculate the first differences: \[ P(0) = 0^2 = 0 \ P(1) = 1^2 = 1 \ P(2) = 2^2 = 4 \ P(3) = 3^2 = 9 \] First differences: \[ \Delta P(0) = P(1) - P(0) = 1 - 0 = 1 \ \Delta P(1) = P(2) - P(1) = 4 - 1 = 3 \ \Delta P(2) = P(3) - P(2) = 9 - 4 = 5 \]
Thus, the first differences are \(1, 3, 5\).
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Calculate the second differences: \[ \Delta^2 P(0) = \Delta P(1) - \Delta P(0) = 3 - 1 = 2 \ \Delta^2 P(1) = \Delta P(2) - \Delta P(1) = 5 - 3 = 2 \]
Thus, the second differences are constant at \(2\).
Therefore, the terms of any quadratic polynomial sequence will always have constant second differences, confirming that quadratic sequences exhibit this property.
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