If $S_r = \begin{vmatrix} 2r & x & n(n+1) \\ 6r^2-1 & y & n^2(2n+3) \\ 4r^3-2nr & z & n^3(n+1) \end{vmatrix}$, then $\sum_{r=1}^n S_r$ does not depend on -
Step-by-Step Solution
Key Concept: The determinant is linear with respect to the columns. By applying the property of summation to the columns, we can evaluate the sum of the determinant.
The determinant $S_r$ can be written as a sum of determinants by linearity. Since the second and third columns do not depend on $r$, the summation $\sum_{r=1}^n S_r$ will involve $\sum r$, $\sum r^2$, and $\sum r^3$. Upon evaluating these sums and simplifying the determinant, it is found that the expression is independent of $n$.
Correct Answer: 3