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  4. If Dr=|(2/2 r-1) 3 0 (1/2 r+1)| then displaystyle∑r=1n Dr equals

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Q. If $D_r=\begin{vmatrix}\frac{2}{2 r-1} & 3 \\ 0 & \frac{1}{2 r+1}\end{vmatrix}$ then $\displaystyle\sum_{r=1}^n D_r$ equals

Determinants

Solution:

$ D_r=\frac{2}{(2 r-1)(2 r+1)}=\left(\frac{1}{2 r-1}-\frac{1}{2 r+1}\right)$
$\displaystyle\sum_{ r =1}^{ n } D _{ r }=1-\frac{1}{2 n +1}=\frac{2 n }{2 n +1}$