Let $D_r =\begin{vmatrix}a&2^r&2^{16}-1\\b &3(4^r)&2(14^{16} -1)\\c&7(8^r)&4(8^{16}-1)\end{vmatrix}$, then the value of $\sum\limits_{k=1}^{16}D_k$, is

0

We have,

$\sum\limits_{k=1}^{16}D_k=\begin{vmatrix}a&\sum\limits_{k=1}^{16}2^k&2^{16}-1\\b &3(\sum\limits_{k=1}^{16}4^k)&2(14^{16} -1)\\c&7(\sum\limits_{k=1}^{16}8^k)&4(8^{16}-1)\end{vmatrix}$

$⇒\sum\limits_{k=1}^{16}D_k=\begin{vmatrix}a&2(\frac{2^{16}-1}{2-1})&2^{16}-1\\b &3×4(\frac{4^{16}-1}{4-1})&2(14^{16} -1)\\c&7×8(\frac{8^{16}-1}{8-1})&4(8^{16}-1)\end{vmatrix}$

$⇒\sum\limits_{k=1}^{16}D_k=\begin{vmatrix}a&2(2^{16}-1)&(2^{16}-1)\\b &4(4^{16}-1)&2(14^{16} -1)\\c&8(8^{16}-1)&4(8^{16}-1)\end{vmatrix}$

$⇒\sum\limits_{k=1}^{16}D_k=2\begin{vmatrix}a&(2^{16}-1)&(2^{16}-1)\\b &2(4^{16}-1)&2(14^{16} -1)\\c&4(8^{16}-1)&4(8^{16}-1)\end{vmatrix}$   [Taking 2 common from $C_2$]

$⇒\sum\limits_{k=1}^{16}D_k=2×0=0$