1. Given the number $a=1+3+5+\ldots+4025+2013$.

a) Calculate $[\sqrt{a}]$ where $[x]$ represents the integer part of the number $x$.

b) Prove the inequality:

$$
\sqrt{a+\sqrt{a+\sqrt{a}}}<2014
$$

Prof. Ana Popescu
## Subject 1.

Given the number $a=1+3+5+\ldots+4025+2013$.

a) Calculate $[\sqrt{a}]$ where $[x]$ represents the integer part of the number $x$.

b) Prove the inequality:

$\sqrt{a+\sqrt{a+\sqrt{a}}}<2014$

a) We have $a=1+3+5+\ldots+4025+2013=2013^{2}+2013=2013 \cdot 2014$

$2013^{2}<2013 \cdot 2014<2014^{2}$, hence $2013<\sqrt{2013 \cdot 2014}<2014$ so $[\sqrt{a}]=2013$... 2p

b) From a) we have that $\sqrt{a}<2014$ $\mathbf{1 p}$

$a+\sqrt{a}<2013 \cdot 2014+2014 \Leftrightarrow a+\sqrt{a}<2014^{2} \Leftrightarrow \sqrt{a+\sqrt{a}}<2014$

Similarly, it follows that $\sqrt{a+\sqrt{a+\sqrt{a}}}<2014$.