Question

Let $D_r=\left[\begin{array}{ccc}a& 2^r& 2^{16}\\ b& 3(4^4)& 2(14^{16}-1)\\ c& 7(8^r)& 4(18^{16}-1)\end{array}\right]$ then the value of $\sum _{r=1}^{16}{D}_r$ is

• A. 0
• B. a + b + c
• C. ab + bc + ca
• D. none of these.

Answer 1

Answer: (A)

$\sum _{r=1}^{16}{D}_r=\left|\begin{array}{ccc}a& \sum _{r=1}^{16}{2}^r& 2^{16}-1\\ b& 3\sum _{r=1}^{16}{4}^r& 2\left(4^{16}-1\right)\\ c& 7\sum _{r=1}^{16}{8}^K& 4\left(8^{16}-1\right)\end{array}\right|$
=$\left|\begin{array}{ccc}a& 2\left(\frac{2^{16}-1}{2-1}\right)& 2^{16}-1\\ b& 3\times 4\left(\frac{4^{16}-1}{4-1}\right)& 2\left(4^{16}-1\right)\\ c& 7\times 8\left(\frac{8^{16}-1}{8-1}\right)& 4\left(8^{16}-1\right)\end{array}\right|$
=$\left|\begin{array}{ccc}a& 2(2^{16}-1)& 2^{16}-1\\ b& 4\left(4^{16}-1\right)& 2\left(4^{16}-1\right)\\ c& 8\left(8^{16}-1\right)& 4\left(8^{16}-1\right)\end{array}\right|$
=2$\left|\begin{array}{ccc}a& 2^{16}-1& 2^{16}-1\\ b& 2\left(4^{16}-1\right)& 2\left(4^{16}-1\right)\\ c& 4\left(8^{16}-1\right)& 4\left(8^{16}-1\right)\end{array}\right|=2\cdot 0=0$