## SUBJECT 2

Given the numbers $a=\sqrt{3}+\sqrt{3^{2}}+\sqrt{3^{3}}+$ $\qquad$ $+\sqrt{3^{2016}}$ and $b=3+3^{2}+3^{3}+$ $\qquad$ $+3^{2016}$

a) Calculate the number $a$.

b) If $c=\frac{b}{a} \cdot \frac{\sqrt{3}+1}{\sqrt{3}}-1$, show that the number $c$ is a perfect square.
## SUBJECT 2

a) Calculation of the number $a=\sqrt{3}+\sqrt{3^{2}}+\sqrt{3^{3}}+\ldots \ldots \ldots .+\sqrt{3^{2016}} \Rightarrow$

$$
a=\sqrt{3}+3+\ldots .+3^{1007} \sqrt{3}+3^{1008}=\sqrt{3}(1+\sqrt{3})\left(1+3+\ldots .+3^{1007}\right) \quad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots
$$

b) Calculation of $b=3+3^{2}+3^{3}+$ $\qquad$ $+3^{2016}=3 \cdot\left(1+3^{1008}\right) \cdot(1+3+$ $\qquad$ $\left.3^{1007}\right)$ $\qquad$ $c=\frac{b}{a} \cdot \frac{\sqrt{3}+1}{\sqrt{3}}-1=3^{1008}$ $\qquad$ $.2 \mathbf{p} ;$ Completion, $c=\left(3^{504}\right)^{2}$ $\qquad$