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The $F_{3}^{-}$ ion has not been observed because fluorine is the most electronegative element in the periodic table, and the formation of the $F_{3}^{-}$ ion would be highly unstable and energetically unfavorable.
The $Br_{3}^{-}$ and $I_{3}^{-}$ ions are known to exist because bromine and iodine are less electronegative than fluorine, and the formation of these ions is more stable and energetically favorable.
The high electronegativity of fluorine makes it difficult to accommodate three fluorine atoms around a central fluoride ion, as the repulsive forces between the highly electronegative fluorine atoms would be too strong. This results in the $F_{3}^{-}$ ion being highly unstable and not observed experimentally.
