#### Answer

Converges to $4$

#### Work Step by Step

$a_{n+1} = \frac{1}{2}\sqrt a_n + 3$; $a_0 = 1000$
*Decimals are approximations
$a_0 = 1000$
$a_1 = 18.8114$
$a_2 = 5.1686$
$a_3 = 4.13673$
$a_4 = 4.01695$
$a_5 = 4.00212$
$a_6 = 4.00026$
$a_7 = 4.00003$
$a_8 = 4.0$
$a_9 = 4.0$
$a_{10} = 4.0$
The terms seem to approach closer and closer to 4 as the sequence continues.