Question
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 ____
JEE Advanced
Mathematics
Sequences and Series
2019
JEE Advanced 2019 (Paper 1)
JEE Advanced Mathematics Question (2019) — Solution
Options
- A. A
- B. B
- C. C
- D. D
Answer
A. A
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 = 4,11 , … When n = 11 , m = 18 and r = 8 ⇒ For 1 s t common term of three series A ( a , d ) a = 1 s t term of A ( 1,3 ) a = 1 + ( 18 1 ) 3 = 52 ⇒ Common difference of A a , d = L C M of common difference of A 1 ; 3 , A 2 ; 5 , A 3 ; 7 d = L C M   ( 3 ,   5 ,   7 ) d = 105 Hence, a + d = 52 + 105 = 157
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