Mathematica, 31 bytes

Nest[{+##/2,(1##)^.5}&@@#&,##]&

Usage:

In[1]:= Nest[{+##/2,(1##)^.5}&@@#&,##]&[{24, 6}, 5]

Out[1]= {13.4582, 13.4582}

Mathematica, 31 30 bytes

Saved one byte thanks to Martin Büttner.

{+##/2,(1##)^.5}&@@#&~Nest~##&

Usage:

In[1]:= {+##/2,(1##)^.5}&@@#&~Nest~##&[{24, 6}, 5]

Out[1]= {13.4582, 13.4582}