A3. Dana je neskončna geometrijska vrsta $\frac{1}{2^{x}}+\frac{1}{2^{2 x}}+\frac{1}{2^{3 x}}+\cdots$. Za katero realno število $x$ je vsota vrste enaka $\frac{1}{7}$ ?
(A) $x=1$
(B) $x=2$
(C) $x=\frac{3}{2}$
(D) $x=3$
(E) $x=0$

A3. The given infinite geometric series is $\frac{1}{2^{x}}+\frac{1}{2^{2 x}}+\frac{1}{2^{3 x}}+\cdots$. For which real number $x$ is the sum of the series equal to $\frac{1}{7}$?
(A) $x=1$
(B) $x=2$
(C) $x=\frac{3}{2}$
(D) $x=3$
(E) $x=0$
A3. We can write the series as $\frac{1}{2^{x}}\left(1+\frac{1}{2^{x}}+\left(\frac{1}{2^{x}}\right)^{2}+\ldots\right)=\frac{1}{2^{x}} \cdot \frac{1}{1-\frac{1}{2^{x}}}=\frac{1}{2^{x}-1}$. From $\frac{1}{7}=\frac{1}{2^{x}-1}$ it follows that $x=3$.