Let -a -a-1 n 6 a-12 2n2 4n-2 a-13 3n3 3n2-3n-the value of -a-1n-a is

Given, Δ_a= a 1 amp;n amp;6 (a 1)^2 amp;2n^2 amp;4n 2 (a 1)^3 amp;3n^3 amp;3n^2 3n ∴ ∑_a=1^n Δ_a = ∑_a=1^n(a 1) amp;n amp;6 ∑_ ...

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Properties of Determinants

Let Δa =a-1 ...

Question

Let $Δ_{a} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a - 1 & n & 6 (a - 1)^{2} & 2 n^{2} & 4 n - 2 (a - 1)^{3} & 3 n^{3} & 3 n^{2} - 3 n \end{matrix} ∣ ∣ ∣ ∣ ∣$ ;
the value of $\sum_{a = 1}^{n} Δ_{a}$ is

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n^{2}

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△ 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

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}

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} ∣ ∣ ∣ ∣ ∣ .

\sum_{n = 1}^{n} 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_{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}

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