Equal in distribution, linear combination of random variables

Use CLT. Let $X_1,X_2,\ldots$ is a sequence of iid random variables with the same distribution as $X$, $S_n=X_1+\ldots+X_n$. Then $$ \frac{S_3}{\sqrt{3}} \overset{d}{=} X, $$ $$ \frac{S_9}{\sqrt{9}} = \frac{\frac{X_1+X_2+X_3}{\sqrt{3}}+\frac{X_4+X_5+X_6}{\sqrt{3}}+\frac{X_7+X_8+X_9}{\sqrt{3}}}{\sqrt{3}} \overset{d}{=} X, $$ and so on. So, for any $n$, $$ \frac{S_{3^n}}{\sqrt{3^n}} \overset{d}{=} X. $$ CLT implies that the distribution of l.h.s. converges to standard normal. Since it coincides with the distribution of $X$, then $X\sim N(0,1)$.