If $a=1+2+4+.........$ to n terms

$b = 1 + 3 + 9 + ......$ to n terms

$c= 1 + 5 + 25+ ........$ to n terms

then $\begin{vmatrix}a & 2b & 4c\\2 & 2 & 2\\2^n & 3^n & 5^n\end{vmatrix}=$

0

The correct answer is option (3) : 0

We have,

$a=1+2+4+.........$ to n terms $=2^n-1$

$b = 1 + 3 + 9 + ......$ to n terms $=\frac{1}{2}(3^n -1)$

$c= 1 + 5 + 25+ ........$ to n terms $=\frac{1}{4}(5^n -1)$

$∴\begin{vmatrix}a & 2b & 4c\\2 & 2 & 2\\2^n & 3^n & 5^n\end{vmatrix}$

$=\begin{vmatrix}2^n-1 & 3^n-1 & 5^n-1\\2 & 2 & 2\\2^n & 3^n & 5^n\end{vmatrix}$

$=2\begin{vmatrix}2^n-1 & 3^n-1 & 5^n-1\\1 & 1 & 1\\1 & 1 & 1\end{vmatrix}$ Applying $R_3→R_3-R_1$ and taking 2 common $R_2$

$=2 ×0= 0 $