Let A = 1,2,3, , 10 and R be a relation on A such that R = ( a , b ): a =2 ~b +1 . Let ( a ₁, a ₂ ) , (a₂, a₃ ), (a₃, a₄ ), ., (a_k, a_ k+1 ) be a sequence of k elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k , for which such a sequence exists, is equal to :

Step-by-step solution

aligned & a =2 ~b +1 & 2 ~b = a -1 & R = (3,1),(5,2), ,(99,49) aligned Let (2 m+1, m),(2 -1, ) are such ordered pairs. According to the condition m =2 -1 ~m = odd number 1^ st element of ordered pair ( a , b ) a=2(2 -1)+1=4 -1 Hence a 3,7, , 99 1,2, , 25 set of sequence aligned & (4 -1,2 -1),(2 -1, -1), ( -1, -2 2 ), . & 2^ nd element of each ordered pair = -2^ r -2 2^ r -2 aligned For maximum number of ordered pairs in such sequence aligned & -2^ r -2 2^ r -2 =1 or 2 ; 1 25 & =2^ r -1 or =3.2^ r -2 aligned aligned