Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If $AP(1;3) \cap AP(2;5)\cap AP(3;7)=AP(a;d),$, then a + d Equals ……….

Now, let  $m^{th}$ term of first progression 
             AP (1; 3) = 1+ (m -1)3 = 3m -2 …..(i) 
            And $n^{th}$  term of second progression 
            AP (2; 5) = 2+ (n -1)5 = 5m -3 …..(ii) 
           And $r^{th}$  term of third progression 
           AP (3; 7) = 3+ (r -1)7 = 7m -4 …..(iii)
          Then, 3m -2 = 5n -3 = 7r -4
          Now, for AP(1; 3) $\cap$  AP(2; 5) $\cap$ AP(3; 7), 
          The common terms of first and second progressions, 
           $m=\frac{5n-1}{3}$
           n = 2, 5, 11,….
         And the common terms of second and the third progressions, 
        $r=\frac{5n+1}{7}\Rightarrow$ n = 4, 11,….
         Now, the first common term of first, second and third progressions is obtained when
         n = 11