An Infinite Number Of Xs

Let $7=S$ .Then: $S=\frac{X}{11}+\frac{X}{121}+\frac{X}{1331}+\frac{X}{14641}+\dotsm$ $\frac{S}{11}=\frac{X}{121}+\frac{X}{1331}+\frac{X}{14641}+\dotsm$ $S-\frac{S}{11}=\left(\frac{X}{11}+\frac{X}{121}+\dotsm\right)-\left(\frac{X}{121}+\frac{X}{1331}+\dotsm\right)$ $\frac{10S}{11}=\frac{X}{11}$ $10S=X$ Plugging in $S=7$ ,We get: $10(7)=X=\boxed{70}$

$7=\frac{X}{11^1}+\frac{X}{11^2}+\frac{X}{11^3}+\frac{X}{11^4}+\cdots \rightarrow n =1, 7 = \frac{X}{11^n}+\frac{X}{11^{n+1}}+\frac{X}{11^{n+2}}+\cdots$ . Then lets get a approximation of $7 = \frac{X}{11}$ . Maybe it is $77=X$ . Next, $7=\frac{X}{11^2} \rightarrow 847=X$ . And so on.... Then, let the final number $A$ . Next, $A=\frac{X}{11^{b}}, b \geq 1$ . Then $7 \times 11^{b-1} = (b-1)X+(b-2)X+(b-3)X+\cdots$ . Although we do not know the value of $b$ , but lets keep calculating. In $7 = \frac{X}{11}+\frac{X}{121}+\frac{X}{1331}+\frac{X}{14641}$ , we get $102487 = 1331X + 121X + 11X + X$ . Then $1464X = 102487$ . Lets finish the calculating with $X = \frac{102487}{1464}$ . Lets round it into the integer form. Its approximation is $70$ , so the answer is $\boxed{70}$ .

$\sum_{n = 0}^{\infty}{ar^n} = \frac{a}{1 - r}$

---

$7 = \frac{X}{11} + \frac{X}{121} = \frac{X}{1331} + \frac{X}{14641} + \cdots$

$a = \frac{X}{11},~ r = \frac{1}{11}$

---

$\sum_{n = 0}^{\infty}{\frac{A}{x} \cdot \left(\frac{B}{x}\right)^n} = \frac{A}{x - B}$

---

$A = X,~ B = 1,~ x = 11$

$\sum_{n = 0}^{\infty}{\frac{X}{11} \cdot \left(\frac{1}{11}\right)^n} = \frac{X}{11 - 1} = \frac{X}{10}$

$7 = \frac{X}{10},~ X = 70$

$\large \boxed{X = 70}$

$7=X(\frac{1}{11}+\frac{1}{121}+\frac{1}{1331}+\frac{1}{14641}+\dotsm)$ $s=(\frac{1}{11}+\frac{1}{121}+\frac{1}{1331}+\frac{1}{14641}+\dotsm)$ $11s=1+\frac{1}{11}+\frac{1}{121}+\frac{1}{1331}+\frac{1}{14641}+\dotsm$ $11s=1+s \longrightarrow s=\dfrac{1}{10}$ $7=Xs \longrightarrow 7=\dfrac{X}{10} \longrightarrow X=\boxed{70}$ note that $\sum_{n=1}^\infty \dfrac{1}{S^n}=\dfrac{1}{S-1}$

Quite easy. Just use the formula for infinite sum of a GP.