$a)a_n=\frac{2n+1}{n},g=2$  

1) $\left|\frac{2n+1}{n}-2\right|<\frac{1}{100}$  

$\left|2+\frac{1}{n}-2\right|<\frac{1}{100}$  

$\left|\frac{1}{n}\right|<\frac{1}{100}$  

$\frac{1}{n}<\frac{1}{100}$  

zatem

$n>100$  

stąd:

$n\in \left\{101,102,\dots \right\}$  

 

2)  $\left|\frac{2n+1}{n}-2\right|<\frac{17}{1000}$  

$\left|2+\frac{1}{n}-2\right|<\frac{17}{1000}$  

$\left|\frac{1}{n}\right|<\frac{1}{\frac{1000}{17}}$  

$\frac{1}{n}<\frac{1}{\frac{1000}{17}}$  

zatem

$n>\frac{1000}{17}$  

$n>58\frac{14}{17}$  

stąd:

$n\in \left\{59,60,\dots \right\}$  

 

3)  $\left|\frac{2n+1}{n}-2\right|<\frac{3}{100000}$  

$\left|2+\frac{1}{n}-2\right|<\frac{3}{100000}$  

$\left|\frac{1}{n}\right|<\frac{3}{100000}$  

$\left|\frac{1}{n}\right|<\frac{1}{\frac{100000}{3}}$  

$\frac{1}{n}<\frac{1}{\frac{100000}{3}}$  

zatem

$n>\frac{100000}{3}$  

$n>33333\frac{1}{3}$  

stąd:

$n\in \left\{33334,33335,\dots \right\}$