Answer
See the picture below.
The sequence converges to 2.
Work Step by Step
If we write the first 10 terms, we can make a conjucture of the convergence of the sequence.
$a_n=\frac{1+4n}{2n}$
$a_1=\frac{1+4\times 1}{2\times 1}=2.5$
$a_2=\frac{1+4\times 2}{2\times 2}=2.25$
$a_3=\frac{1+4\times 3}{2\times 3}\approx 2.17$
$a_4=\frac{1+4\times 4}{2\times 4}\approx 2.13$
$a_5=\frac{1+4\times 5}{2\times 5}\approx 2.1$
$a_6=\frac{1+4\times 6}{2\times 6}\approx 2.08$
$a_7=\frac{1+4\times 7}{2\times 7}\approx 2.07$
$a_8=\frac{1+4\times 8}{2\times 8}\approx 2.06$
$a_9=\frac{1+4\times 9}{2\times 9}\approx 2.055$
$a_{10}=\frac{1+4\times 10}{2\times 10}\approx 2.05$
Also, if we look at really huge numbers, for example 10,000 and 100,000 we can see the convergence.
$a_{10000}=\frac{1+4\times 10000}{2\times 10000}\approx 2.00005$
$a_{100000}=\frac{1+4\times 100000}{2\times 100000}=2.000005$
