Let A P a ; d denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0 . If A P 1 ; 3 ∩ A P 2 ; 5 ∩ A P 3 ; 7 = A P a ; d then a + d equals ____

Step-by-step solution

Let T 1 ,   m : m t h term of 1 s t series A 1 ; 3 T 2 ,   n : n t h term of 2 n d series A 2 ; 5 T 3 ,   r : r t h term of 3 r d series A ( 3 ; 7 ) For common terms T ( 1 ,   m ) = T ( 2 ,   n ) = T ( 3 ,   r ) 1 + 3 m - 1 = 2 + 5 n - 1 = 3 + 7 r - 1 3 m - 2 = 5 n - 3 = 7 r - 4 ⇒ m = 5 n - 1 3     a n d     r = 5 n + 1 7 For ‘ m ’ to be a natural number n = 2,5 , 11 , … And for ‘ r ’ to be a natural number n = 4,1