Given $2\left( a _1+ a _2+\ldots . .+ a _{ n }\right)= b _1+ b _2+\ldots . .+ b _{ n }$

$\Rightarrow \quad 2 \times \frac{ n }{2}\left(2 c +( n -2) x _2\right)= c \left(\frac{2^{ n }-1}{2-1}\right)$

$\Rightarrow \quad 2 n ^2-2 n = c \left(2^{ n }-1-2 n \right)$

$\Rightarrow \quad c =\frac{2 n ^2-2 n }{2^{ n }-1-2 n } \in N$

$\text { So, }  2 n ^2-2 n \geq 2^{ n }-1-2 n$

$\Rightarrow \quad  2 n ^2+1 \geq 2^{ n } \Rightarrow n <7$

$\Rightarrow \quad  n \text { can be } 1,2,3, \ldots.$

Checking $c$ against these values of $n$

we get $c=12 \quad($ when $n=3)$

Hence number of such $c=1$