To find the 2nd differences of the sequence generated by the polynomial \( n^2 + n + 1 \), we first need to evaluate the polynomial for several values of \( n \) and then calculate the first and second differences.
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Calculate the values of the polynomial: \[ \begin{align*} n = 1: & \quad 1^2 + 1 + 1 = 3 \ n = 2: & \quad 2^2 + 2 + 1 = 7 \ n = 3: & \quad 3^2 + 3 + 1 = 13 \ n = 4: & \quad 4^2 + 4 + 1 = 21 \ n = 5: & \quad 5^2 + 5 + 1 = 31 \ \end{align*} \] So the sequence is: \( 3, 7, 13, 21, 31 \).
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Calculate the first differences: \[ \begin{align*} 7 - 3 & = 4 \ 13 - 7 & = 6 \ 21 - 13 & = 8 \ 31 - 21 & = 10 \ \end{align*} \] The first differences are: \( 4, 6, 8, 10 \).
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Calculate the second differences: \[ \begin{align*} 6 - 4 & = 2 \ 8 - 6 & = 2 \ 10 - 8 & = 2 \ \end{align*} \] The second differences are: \( 2, 2, 2 \).
Thus, the second differences of the sequence generated by the polynomial \( n^2 + n + 1 \) are constant and equal to \( 2 \).