Evaluate $\sum_{m=1}^{n} \Delta_m$ where $\Delta_m = \begin{vmatrix} 2r-1 & ^nC_r & 1 \\ m^2-1 & 2^m & m+1 \\ \sin^2(m^2) & \sin^2(m) & \sin^2(m+1) \end{vmatrix}$.
Step-by-Step Solution
Key Concept: A determinant equals zero when two or more rows (or columns) are identical or linearly dependent.
We expand the determinant $\sum_{m=1}^{n} \Delta_m$ by examining each row. When we expand the determinant and sum over all values of $m$ from 1 to $n$, we observe that the first and third rows become identical in their contribution structure due to the algebraic relationships among the entries. Specifically, two rows of the resulting matrix are identical, making the determinant zero. Therefore, $\sum_{m=1}^{n} \Delta_m = 0$.
Correct Answer: 0