Question

Let $D_k=\left|\begin{array}{ccc}a& 2^k& 2^{16}-1\\ b& 3\left(4^k\right)& 2\left(4^{16}-1\right)\\ c& 7\left(8^k\right)& 4\left(8^{16}-1\right)\end{array}\right|$, then the value of $\sum _{k=1}^{16}{D}_k$ is

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

Answer 1

Answer: (A)

We have, $\sum _{k=1}^{16}{D}_k=\left|\begin{array}{ccc}a& \left(\sum _{k=1}^{16}{2}^k\right)& 2^{16}-1\\ b& 3\left(\sum _{k=1}^{16}{4}^4\right)& 2(4^{16}-1)\\ c& 7\left(\sum _{k=1}^{16}{8}^k\right)& 4(8^{16}-1)\end{array}\right|$
$\Rightarrow$ $\sum _{k=1}^{16}{D}_k=\left|\begin{array}{ccc}a& \left(\frac{2^{16}-1}{2-1}\right)& 2^{16}-1\\ b& 3\times 4\left(\frac{4^{16}-1}{4-1}\right)& 2(4^{16}-1)\\ c& 7\times 8\left(\frac{8^{16-1}}{8-1}\right)& 4(8^{16}-1)\end{array}\right|$
$\Rightarrow$ $\sum _{k=1}^{16}{D}_k=\left|\begin{array}{ccc}a& 2(2^{16}-1)& 2^{16}-1\\ b& 4(4^{16}-1)& 2(4^{16}-1)\\ c& 8(8^{16}-1)& 4(8^{16}-1)\end{array}\right|$
$\Rightarrow$ $\sum _{k=1}^{16}{D}_k=2\left|\begin{array}{ccc}a& (2^{16}-1)& 2^{16}-1\\ b& 2(4^{16}-1)& 2(4^{16}-1)\\ c& 4(8^{16}-1)& 4(8^{16}-1)\end{array}\right|$
[taking $2$ common from $C_2$ ]
$\Rightarrow$ $\sum _{k=1}^{16}{D}_k=2\times 0=0$