Let a1,a2,a3,⋯ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b1,b2,b3,⋯ be a sequence of positive integers in geometric progression with common ratio 2. If a1=b1=c, then the number of all possible values of c, for which the equality2(a1+a2+⋯+an)=b1+b2+⋯+bnholds for some positive integer n, is

Let a1,a2,a3,⋯ be a sequence of positive integers in arithmetic prog

1.	Let a1,a2,a3,⋯ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b1,b2,b3,⋯ be a sequence of positive integers in geometric progression with common ratio 2. If a1=b1=c, then the number of all possible values of c, for which the equality2(a1+a2+⋯+an)=b1+b2+⋯+bnholds for some positive integer n, is
Answer» Let $a_{1}, a_{2}, a_{3}, \dots$ be a sequence of positive integers in arithmetic progression with common difference $2.$ Also, let $b_{1}, b_{2}, b_{3}, \dots$ 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 $2 (a_{1} + a_{2} + \dots + a_{n}) = b_{1} + b_{2} + \dots + b_{n}$ holds for some positive integer $n,$ is