a) Verify that $(x-1)(x+1)+1=x^{2}$.

b) Find the value of $\sqrt{1+2014 \sqrt{1+2015 \sqrt{1+2016 \cdot 2018}}}$.
Solution

a) $(x-1)(x+1)+1=\left(x^{2}-1\right)+1=x^{2}$.

b) Using the previous item several times, we obtain:

$$
\begin{aligned}
\sqrt{1+2014 \sqrt{1+2015 \sqrt{1+2016 \cdot 2018}}} & = \\
\sqrt{1+2014 \sqrt{1+2015 \sqrt{2017^{2}}}} & = \\
\sqrt{1+2014 \sqrt{1+2015 \cdot 2017}} & = \\
\sqrt{1+2014 \sqrt{2016^{2}}} & = \\
\sqrt{1+2014 \cdot 2016} & = \\
\sqrt{2015^{2}} & = \\
2015 &
\end{aligned}
$$