If D_{n}=\begin{vmatrix} n & 20 & 30 \\ n^{2} & 40 & 50 \\ n^{3} & 60 & 70 \end{vmatrix} then what is the value of \sum_{n=1}^{4}D_{n}?

Explanation

Because the summation only affects the first column, we first calculate \sum n = 10, \sum n^2 = 30, and \sum n^3 = 100. Replacing the first column with these sums gives the determinant \begin{vmatrix} 10 & 20 & 30 \\ 30 & 40 & 50 \\ 100 & 60 & 70 \end{vmatrix}. Taking 10^3 out as a common factor, we evaluate \begin{vmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 10 & 6 & 7 \end{vmatrix} = 1(28-30) - 2(21-50) + 3(18-40) = -2 + 58 - 66 = -10. Multiplying by 1000 yields -10000.

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