Cubic Pattern Formula

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$\begin{aligned} a_n&=An^3+Bn^2+Cn+D \\ \\ \Rightarrow\quad \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix} &=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 2^3 & 2^2 & 2 & 1 \\ 3^3 & 3^2 & 3 & 1 \\ 4^3 & 4^2 & 4 & 1 \end{pmatrix} \begin{pmatrix} A \\ B \\ C \\ D \end{pmatrix} \\ \Rightarrow\quad \begin{pmatrix} A \\ B \\ C \\ D \end{pmatrix} &=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 2^3 & 2^2 & 2 & 1 \\ 3^3 & 3^2 & 3 & 1 \\ 4^3 & 4^2 & 4 & 1 \end{pmatrix}^{-1} \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix} \\ &=\frac16\begin{pmatrix}-1&3&-3&1\\9&-24&21&-6\\-26&57&-42&11\\24&-36&24&-6\end{pmatrix} \begin{pmatrix} 3 \\ 10 \\ 29 \\ 66 \end{pmatrix} \\ &=\begin{pmatrix} 1 \\ 0 \\ 0 \\ 2 \end{pmatrix} \quad\checkmark \end{aligned}$

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$\begin{aligned} a_n&=\frac{d}{6}n^3+\left(\frac{a_1-2a_2+a_3}{2}-d\right)n^2+\left(\frac{11d}{6}-\frac{5a_1-8a_2+3a_3}{2}\right)n+\left(3a_1-3a_2+a_3-d\right) \\ &=3n^3+\left(\frac{17-118+143}{2}-18\right)n^2+\left(33-\frac{85-472+429}{2}\right)n+\left(51-177+143-18\right) \\ &=3n^3+3n^2+12n-1 \end{aligned}$

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$\begin{array}{ccccccccccccc} 17 & & 59 & & 143 & & 287 & & 509 & & 827 & & \cdots \\ \\ & 42 & & 84 & & 144 & & 222 & & 318 & & \cdots & \\ \\ & & 42 & & 60 & & 78 & & 86 & & \cdots & & \\ \\ &&& 18 && 18 && 18 && \cdots && \end{array}$

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Cubic Pattern Formula

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