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 $2(a_1 + a_2 + \ldots + a_n) = b_1 + b_2 + \ldots + b_n$ holds for some positive integer $n$, is ______

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 equality 2(a₁ + a₂ + … + aₙ) = b₁ + b₂ + … + bₙ holds for some positive integer n, is ______

$\text{Given: } a_1, a_2, a_3, \ldots \text{ AP, common difference } d = 2$

$b_1, b_2, b_3, \ldots \text{ GP, common ratio } r = 2$

$a_1 = b_1 = c$

$\text{To find: number of possible values of } c \text{ such that}$

$2(a_1 + a_2 + \cdots + a_n) = b_1 + b_2 + \cdots + b_n$

$\text{for some positive integer } n$

$a_n = c + (n-1)\cdot 2 = c + 2(n-1)$

$b_n = c\cdot 2^{\,n-1}$

$S_A = \dfrac{n}{2}\,[2a_1 + (n-1)d]$

$S_A = \dfrac{n}{2}\,[2c + (n-1)\cdot 2] = n(c + n - 1)$

$S_B = \dfrac{b_1(r^n - 1)}{r - 1}$

$S_B = \dfrac{c(2^n - 1)}{2 - 1} = c(2^n - 1)$

$2 S_A = S_B$

$2n(c + n - 1) = c(2^n - 1)$

$2nc + 2n(n-1) = c\,2^n - c$

$c(2^n - 1 - 2n) = 2n(n-1)$

$c = \dfrac{2n(n-1)}{2^n - 2n - 1}$

$\text{Need } c > 0 \Rightarrow 2^n - 2n - 1 > 0$

$\Rightarrow n \ge 3 \quad (n = 1, 2 \text{ give } -1)$

$c \ge 1 \Rightarrow 2^n - 2n - 1 \le 2n(n-1)$

$2^n \le 2n^2 + 1 \Rightarrow n \le 6$

$\text{so } n \in \{3, 4, 5, 6\}$

$c(3) = \dfrac{2\cdot 3\cdot 2}{2^3 - 6 - 1} = \dfrac{12}{1} = 12$

$c(4) = \dfrac{2\cdot 4\cdot 3}{2^4 - 8 - 1} = \dfrac{24}{7}$

$c(5) = \dfrac{2\cdot 5\cdot 4}{2^5 - 10 - 1} = \dfrac{40}{21}$

$c(6) = \dfrac{2\cdot 6\cdot 5}{2^6 - 12 - 1} = \dfrac{60}{51} = \dfrac{20}{17}$

$\text{Only } c = 12 \text{ is an integer } \left(\tfrac{24}{7}, \tfrac{40}{21}, \tfrac{20}{17}\right)$

$a: 12, 14, 16 \Rightarrow S_A = 42,\quad 2S_A = 84$

$b: 12, 24, 48 \Rightarrow S_B = 84$

$\therefore 2S_A = S_B \ \checkmark$

$\boxed{\text{Number of possible values of } c = 1}$