Question

If a1,a2, a3, ........ , an and b1, b2, b3, ........ ,bn are two Arithmetic Progressions.Is a1 + b1, a2 + b2, ........, an + bn an Arithmetic Progression? Why/ Why not?

Answer 1

let a1 be 1
     a2 be 2 
     a3 be 3 and so on
so the arithemetic progression will be 1,2,3,4,....
let b1 be 2
    b2 be 2
    b3 be 2 and so on
so the arithemetic progression will be 2,2,2,2,...
a1+ b1 = 1+2 =3
a2+ b2 = 2+2 =4
a3 + b3 = 3+2 =5
a4 + b4 =4+2 =6 and so on
so the arithemietic progression will be 3,4,5,6....
to verify it 4-3 should be equal to 5-4 which is equal to 1
so it is an arithemetic progression

Answer 2

Yes, (a1+b1), (a2+b2), (a3+b3),........,(an+bn) is in AP

Let Sequence a1, a2, a3,....an have common difference be (d1)
And that of b1, b2, b3,....bn be d2

According to condition:-
Sequence (a1+b1), (a2+b2), (a3+b3),........,(an+bn),
Now the each succeding terms of this sequence differ by d1+ d2 which is constant for all the terms of this AP. So it is an AP.

For eq:-
Let A = a1, a2, a3,....an = 2,4,6,8....(here d = 2)
Let B = b1, b2, b3,....bn = 1,3,5,7,....(here also d = 2)
Now sequence (a1+b1), (a2+b2), (a3+b3),........,(an+bn) = (2+1), (4+3), (6+5), (8+7) = 3,7,11,15......
In this sequence, 7-3=4, 11-7 = 4, 15-11 = 4. So it is an AP.....!