Question

which polynomial sequence shows that the terms of the 2nd differences are constant

Answer 1

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:

1. **Calculate the first differences** of the sequence generated by the polynomial:
\[
\Delta P(n) = P(n+1) - P(n)
\]

2. **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 \( P(n) = n^2 \):

1. **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 \(1, 3, 5\).

2. **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.