If `D_(r) = |(r,1,(n(n +1))/(2)),(2r -1,4,n^(2)),(2^(r -1),5,2^(n) -1)|`, then the value of `sum_(r=1)^(n) D_(r)`, is

Options

A. 0
B. 1
C. `(n(n +1) (2n+1))/(6)`
D. none of these

Answer

A

Explanation

We have,
`underset(r=0)overset(n)sum D_(r) = |(underset(r =1)overset(n)sum r,1,(n(n +1))/(2)),(underset(r=1)overset(n)sum 2r -1,4,n^(2)),(underset(r =1)overset(n)sum 2^(r -1),5,2^(n) -1)|`
`rArr underset(r =0)overset(n)sum D_(r) = |((n(n +1))/(2),1,(n(n+1))/(2)),(n^(2),4,n^(2)),(2^(n) -1,5,2^(n) -1)| =0`