If S= a R :|2 a-1|=3[a]+2 a , where [t] denotes the greatest integer less than or equal to t and t represents the fractional part of t , then 72 _ a S a is equal to ______

Step-by-step solution

aligned & |2 a -1|=3[ a ]+2 a & |2 a -1|=[ a ]+2 a aligned aligned & Case-1 : a > 1 2 & 2 a-1=[a]+2 a & [a]=-1 a [-1,0) Reject & Case-2 : a < 1 2 & -2 a +1=[ a ]+2 a & a = I + f & -2( I + f )+1= I +2 I +2 f & I =0, f = 1 4 a = 1 4 & Hence a = 1 4 & 72 _ a S a =72 1 4 =18 aligned