If - r - - - r - 1 n 6- r - 1 2 2 n 2 4 n - 2- r - 1 3 3 n 3 3 n 2 - 3 n - - then - n - 1 n - r is

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Matrix Definition and Representation

If Δ r = ...

Question

If $Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} r - 1 & n & 6 (r - 1)^{2} & 2 n^{2} & 4 n - 2 (r - 1)^{3} & 3 n^{3} & 3 n^{2} - 3 n \end{matrix} ∣ ∣ ∣ ∣ ∣ .$ then $\sum_{n = 1}^{n} Δ_{r}$ is

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Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} r - 1 & n & 6 (r - 1)^{2} & 2 n^{2} & 4 n - 2 (r - 1)^{3} & 3 n^{3} & 3 n^{2} - 3 n \end{matrix} ∣ ∣ ∣ ∣ ∣

n \sum r = 1 Δ_{r}

D_{r} =

∣ ∣ ∣ ∣ ∣ \begin{matrix} r - 1 & n & 6 {(r - 1)}^{2} & 2 n^{2} & 4 n - 2 {(r - 1)}^{3} & 3 n^{3} & 3 n^{2} - 3 n \end{matrix} ∣ ∣ ∣ ∣ ∣

n \sum r = 1 D_{r} =

Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} r & n & 6 r^{2} & {2 n}^{2} & 4 n - 2 r^{3} & {3 n}^{3} & 3 n^{2} - 3 n \end{matrix} ∣ ∣ ∣ ∣ ∣

n - 1 \sum r = 0 Δ_{r}

Δ_{a} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a - 1 & n & 6 (a - 1)^{2} & 2 n^{2} & 4 n - 2 (a - 1)^{3} & 3 n^{3} & 3 n^{2} - 3 n \end{matrix} ∣ ∣ ∣ ∣ ∣

\sum_{a = 1}^{n} Δ_{a}

△ a = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a - 1 & n & 6 {(a - 1)}^{2} & 2 n^{2} & 4 n - 2 {(a - 1)}^{3} & 3 n^{3} & 3 n^{2} - 3 n \end{matrix} ∣ ∣ ∣ ∣ ∣

\sum_{a - 1}^{n} △ a

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