When the corresponding terms of two AP are added are the terms of the resulting series also an AP?

Yes.

If two series which are
in arithmetic progression are added, term by term  correspondingly, then the resulting
 series is also in
AP.

Proof:

  Let  a1 ,  d, and
n be the first term , the common difference and the number of terms  in a series series
A  of an AP.

Then the nth term an of the series A  is
giveny by the relation:

 an =
a1+(n-1)d

Similarly let , b1 , e  and n be the first term,
the common difference  and the number of the terms of another series
B.

Then the nth term bn of the series B is given by:
b1+(n-1)e.

Clearly , during the the addition of the
corresponding terms nth terms of the series A and B is : an+bn = a1+(n-1)d
}+{(b1+(n-1)e)} = (a1+b1) + (n-1)(d+e). Or

The nth term 
ahter adding the corresponding term bu terms of both series is given by (an+bn) =
(a1+b1) +(n-1)(d+e)  = k1+(n-1)f , where f = d+e.

Thus the
new series is  also an AP with starting term  a1+b1 = k1 and a common ratio d+e =
f.