Kumar sustained the injury during India's series-clinching win in Kolkata, according to an Indian cricket board release.
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The 26-year-old Kumar played a key role in India's 178-run win in the second Test after taking six wickets in the match including a five-wicket haul in the first innings.
Kumar's injury is a double blow to India's seam attack after Ishant Sharma was ruled out with the mosquito-borne disease chikungunya ahead of the three-match series.
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Medium-pacer Thakur, who is yet to play an international match for India, was part of the team's tour to the West Indies in July-August.
The final Test starts October 8 at Indore's Holkar Cricket Stadium.
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