Let APa;d  denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If  AP1;3∩AP2;5∩AP3;7=APa;d  then a + d equals_

AP(1;3)={1,4,7,10,……….} say S1AP(2;5)={2,7,12,17,……….} say S2AP(3;7)={3,10,17,24,……….} say S3S1∩S2={7,22,37,52,…….} i.e., AP(7;15) To find the term common to AP(7;15) and AP(3;7)¯7+15l=3+7m ∴m=4+l7+2l where l,m∈W⇒l can be 3 ∴m=7    therefore the first common term =7+15l=7+153=52and the second common term  =7+1510=157     since  m=4+l7+2l is ∈W , then l can be =10 therefore common difference =157-52=105 ∴S1∩S2∩S3=A.P(52;105)        ∴a+d=157