If Ar = |(1,r,2^r)(2,n,n^2)(n,n(n+1)/2,2^(n+1)| ,then the value of ∑^n (r=1) Ar is ​

Question

Mark the correct alternative in the following :

If A_r = \(\begin{vmatrix} 1 & r& 2^r \\[0.3em] 2 & n &n^2 \\[0.3em] n & \frac{n(n+1)}{2} & 2^{n+1} \end{vmatrix}\),then the value of \(\displaystyle\sum_{r=1}^{n} A_r\) is

A. n 

B. 2n 

C. -2n 

D. n^2

Answer 1

C. - 2n

\(\displaystyle\sum_{r=1}^{n} A_r\) = \(\begin{vmatrix} 1 & r& 2^r \\[0.3em] 2 & n &n^2 \\[0.3em] n & \frac{n(n+1)}{2} & 2^{n+1} \end{vmatrix}\)

1(4 - 1)-1(8 - 1) + 2(2 - 1) = - 2

Answer = c(- 2n)