Question

Let $D_r=\begin{bmatrix} a& 2^r & 2^{16} \\[0.3em] b& 3(4^4) & 2(14^{16}-1)\\[0.3em] c & 7(8^r)& 4(18^{16}-1) \end{bmatrix}$ then the value of $\displaystyle\sum_{r=1}^{16}D_r$ is

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

Answer 1

Answer: (A)

$\displaystyle\sum_{r=1}^{16} \, D_r = \begin{vmatrix}a&\displaystyle\sum_{r=1}^{16} \, 2^r &2^{16}-1\\ b&3\displaystyle\sum_{r=1}^{16} \, 4^r &2\left(4^{16}-1 \right)\\ c&7\displaystyle\sum_{r=1}^{16} \, 8^K&4\left(8^{16}-1\right)\end{vmatrix}$
=$\begin{vmatrix}a&2 \left( \frac{2^{16} - 1 }{2 - 1} \right) &2^{16}-1\\ b&3 \times 4 \left( \frac{4^{16} - 1 }{4 - 1} \right) &2 \left(4^{16}-1 \right)\\ c&7 \times 8 \left( \frac{8^{16} - 1 }{8 - 1} \right) &4\left(8^{16}-1\right)\end{vmatrix}$
=$\begin{vmatrix}a&2(2^{16}-1) &2^{16}-1\\ b&4 \left(4^{16}-1 \right) &2 \left(4^{16}-1 \right)\\ c&8\left(8^{16}-1\right) &4\left(8^{16}-1\right)\end{vmatrix}$
=2$\begin{vmatrix}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{vmatrix} = 2\cdot0 = 0 $