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 _____

Question

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 ______

TestHub · Accepted Answer

​ ∣2 a − 1∣ = 3 [ a ] + 2 { a } ∣2 a − 1∣ = [ a ] + 2 a ​ ​ Case-1 : a > 2 1 ​ 2 a − 1 = [ a ] + 2 a [ a ] = − 1 ∴ a ∈ [ − 1 , 0 ) Reject Case-2 : a < 2 1 ​ − 2 a + 1 = [ a ] + 2 a a = I + f − 2 ( I + f ) + 1 = I + 2 I + 2 f I = 0 , f = 4 1 ​ ∴ a = 4 1 ​ Hence a = 4 1 ​ 72 a ∈ S ∑ ​ a = 72 × 4 1 ​ = 18 ​

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