Let `D_(r) = |(a,2^(r),2^(16) -1),(b,3(4^(r)),2(4^(16) -1)),(c,7(8^(r)),4(8^(16) -1))|`, then the value of `underset(k =1)overset(16)Sigma D_(k)`, is (a) 0 (b) `a+b+c` (c)`ab+bc+ca` (d) 1

Options

A. 0
B. `a+b+c`
C. `ab+bc+ca`
D. `1`

Answer

A

Explanation

`sum_(r=1)^(16)S_(r)=|(a, sum_(r=1)^(16)2^(r),2^(16)-1),(b, 3(sum_(r=1)^(16)4^(r)), 2(4^(16)-1)),(c, 7(sum_(r=1)^(16)8^(r)),4(8^(16)-1))|`
`=|(a, (2(2^(16)-1))/(2-1),2^(16)-1),(b, 3xx4((4^(16)-1))/(4-1), 2(4^(16)-1)),(c, 7xx8((8^(16)-1))/(8-1), 4(8^(16)-1))|`
`=|(a, 2(2^(16)-1),2^(16)-1),(b, 4(4^(16)-1),2(4^(16)-1)),(c,8(8^(16)-1),4(8^(16)-1))|=0`