If `m` is a positive integer and `D_r=|2r-1\ ^m C_r1m^2-1 2^m m+1s in^2(m^2)s in^2(m)s in^2(m+1)|` . Prove that `sum_(r=0)^m D_r=0` .

Explanation

Given,

`D_r=|[2r-1, ^m C_r,1],[m^2-1 ,2^m ,m+1],[sin^2(m^2),sin^2(m),sin^2(m+1)]|`

now,

`sum_(r=0)^m D_r=|[m^2-1, 2^m ,m+1],[m^2-1 ,2^m ,m+],[sin^2(m^2),sin^2(m),sin^2(m+1)]|`

since,`R_1=R_2`

`sum_(r=0)^m D_r=0`