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

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 $R_2$

$=2×0=0$