If $\Delta_k=\left|\begin{array}{ccc}1 & 0 & 0 \ 0 & k & k-1 \ 0 & k-1 & k\end{array} ight|$, then $\Delta_1+\Delta_2+\ldots+\Delta_{20}$ is equal to

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If $\Delta_k=\left|\begin{array}{ccc}1 & 0 & 0 \ 0 & k & k-1 \ 0 & k-1 & k\end{array}ight|$, then $\Delta_1+\Delta_2+\ldots+\Delta_{20}$ is equal to

Aicharya · Accepted Answer

We have, $ \begin{aligned} \Delta_K & =\left|\begin{array}{ccc} 1 & 0 & 0 \ 0 & k & k-1 \ 0 & k-1 & k \end{array}ight| \ \Delta_K & =k^2-(k-1)^2 \ \Delta_1+\Delta_2+\Delta_3 & +\ldots+\Delta_{20} \ 	herefore \quad & =\left(1^2-0^2ight)+\left(2^2-1^2ight)+\ldots\left(20^2-19^2ight) \ & =20^2=400 \end{aligned} $

If $\Delta_k=\left|\begin{array}{ccc}1 & 0 & 0 \\ 0 & k & k-1 \\ 0 & k-1 & k\end{array}\right|$, then…

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