If S=a∈R:∣2a−1∣=3[a]+2a, where [t] denotes the greatest integer less than or equal to t and t represents the fractional part of t, then 72∑a∈Sa is equal to \_\_\_\_\_\_\_\_\_.
Answer: 18
Solutions · 1
Official solution
S:a∈R:∣2a−1∣=3[a]+2a∣2a−1∣=3[a]+2(a−[a])∣2a−1∣=[a]+2a
Case I: If 0\[\begin{gathered} 1-2a=0+2a \\ \Rightarrow a=\frac{1}{4} \end{gathered}\]CaseII:If\frac{1}{2}2a−1=0+2a
No solution
Case III: If 1≤a2a-1=1+2a\RightarrowNosolution\therefore onlysolutionisa=\frac{1}{4}