Answer :
a1, a2, a3, .... an are in AP : let the common difference be d
b1, b2, b3 .... bn are in AP : let the common difference be e.
a1 + b1
a2 + b2 = a1 + b1 + (d + e)
a3 + b3 = a1 + b1 + 2 (d + e)
an + bn = a1 + b1 + (n-1) d + (n-1)e = (a1+b1) + (n-1) (d+e)
So an + bn is an AP with first term a1 + b1 and common difference = d+e.
b1, b2, b3 .... bn are in AP : let the common difference be e.
a1 + b1
a2 + b2 = a1 + b1 + (d + e)
a3 + b3 = a1 + b1 + 2 (d + e)
an + bn = a1 + b1 + (n-1) d + (n-1)e = (a1+b1) + (n-1) (d+e)
So an + bn is an AP with first term a1 + b1 and common difference = d+e.