Question

Let $a_1,a_2,a_3,\cdots$ be a sequence of positive integers in arithmetic progression with common difference $2.$ Also, let $b_1,b_2,b_3,\cdots$ 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
$2(a_1+a_2+\cdots +a_n)=b_1+b_2+\cdots +b_n$
holds for some positive integer $n,$ is

Answer 1

$2(a_1+a_2+\cdots +a_n)=b_1+b_2+\cdots +b_n$
$\Rightarrow 2\left[\frac{n}{2}\right(2a_1+(n-1)2\left)\right]=\frac{b_1(2^n-1)}{2-1}$
$\Rightarrow 2n[a_1+(n-1\left)\right]=b_1(2^n-1)$
$\Rightarrow 2n{a}_1+2n^2-2n=a_1(2^n-1)(\because a_1=b_1)$
$\Rightarrow 2n^2-2n=a_1(2^n-1-2n)$
$a_1=\frac{2(n^2-n)}{(2^n-1-2n)}=c(\because a_1=c)$
$\because c\ge 1$
therefore, $n=1,2,3,4,5,6$
$n=1\Rightarrow c=0\left(\text{rejected}\right)$
$n=2\Rightarrow c<0\left(\text{rejected}\right)$
$n=3\Rightarrow c=12\left(\text{correct}\right)$
$n=4\Rightarrow c=\text{not}\text{Integer}$
$n=5\Rightarrow c=\text{not}\text{Integer}$
$\therefore c=12$ for $n=3$
Hence, no. of such $c=1$