Answer
Converges to $4$
Work Step by Step
$a_{n+1} = \frac{1}{2}a_n + 2$; $a_0 = 3$
*Decimals are approximations
$a_0 = 3$
$a_1 = 3.5$
$a_2 = 3.75$
$a_3 = 3.875$
$a_4 = 3.9375$
$a_5 = 3.96875$
$a_6 = 3.98438$
$a_7 = 3.99219$
$a_8 = 3.99609$
$a_9 = 3.99805$
$a_{10} = 3.99902$
The sequence seems to approach $4$.