Question

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive integers in arithmetic progression with common difference $2$ Also, let $b_{1}, b_{2}, b_{3}, \ldots$ be a sequence of positive integers in geometric progression with common ratio $2$ .If $a_{1}=b_{1}=c$, then the number of all possible values of $c$, for which the equality $2\left(a_{1}+a_{2}+\ldots+a_{n}\right)=b_{1}+b_{2}+\ldots +b_{n}$ holds for some positive integer $n$, is

Answer 1

$\text{Positive integer}$ $n$ is $\underline{1}.$