Answer
See the picture below.
The sequence converges to 0.5.
Work Step by Step
If we write the first 10 terms, we can make a conjucture of the convergence of the sequence.
$a_n=\frac{n+4}{2n}$
$a_1=\frac{1+4}{2\times 1}=2.5$
$a_2=\frac{2+4}{2\times 2}=1.5$
$a_3=\frac{3+4}{2\times 3}\approx 1.17$
$a_4=\frac{4+4}{2\times 4}=1$
$a_5=\frac{5+4}{2\times 5}=0.9$
$a_6=\frac{6+4}{2\times 6}\approx 0.84$
$a_7=\frac{7+4}{2\times 7}\approx 0.79$
$a_8=\frac{8+4}{2\times 8}=0.75$
$a_9=\frac{9+4}{2\times 9}\approx 0.72$
$a_{10}=\frac{10+4}{2\times 10}=0.7$
Also, if we look at really huge numbers, for example 10,000 and 100,000 we can see the convergence.
$a_{10000}=\frac{10000+4}{2\times 10000}=0.5002$
$a_{100000}=\frac{100000+4}{2\times 100000}=0.50002$
