The stabbing took place at a home in Douglasville, west of Atlanta, WSB-TV, Atlanta, reported.
Police said the teen's 77-year-old great-grandmother was fatally stabbed while the grandmother, thought to be in her 50s, was being treated at a hospital for non-life-threatening injuries.
The 14-year-old barricaded himself inside the home for a while, shooting at police with a pellet gun, but was subdued with a Taser and arrested, Douglas County sheriff's Maj. Tommy Wheeler said.
Authorities said police had been called to the home twice in recent months and both times the teen had been hospitalized for evaluation and released.
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