Let A be the sequence of the first 41 terms of an arithmetic progression starting with 2 and having a common difference of 3 . Let B be the sequence of the first 31 terms of an arithmetic progression starting with 2 and having a common difference of 4 . Let c₁ < c₂ < < c_n be the common terms of sequences A and B . If _ i=1 ^ n-1 1 c_i c_ i+1 = p q , where p and q are coprime positive integers, then the value of p +

Step-by-step solution

First, we determine the terms of sequences A and B . Sequence A is an AP with first term a = 2 and common difference d₁ = 3 . Its 41^ st term is 2 + (41 - 1) 3 = 122 . Sequence B is an AP with first term b = 2 and common difference d₂ = 4 . Its 31^ st term is 2 + (31 - 1) 4 = 122 . The common terms of A and B form a new arithmetic progression C . The first common term is c₁ = 2 . The common difference of C is LCM (d₁, d₂) = LCM (3, 4) = 12 . The terms of C are 2, 14, 26, Since both sequences A and B end at 122 , th