Both the seamers had scans two days ago, Broad on his left heel and Finn his lower back, and the results ruled them out of the final Test which began in Nagpur on Thursday. Twenty20 captain Broad will also miss two matches against India in the shortest international format next week, when Eoin Morgan will instead lead the team.
The England and Wales Cricket Board confirmed Broad and Finn's anticipated departures, and announced uncapped Middlesex seamer James Harris is to be added to the Twenty20 squad.
"Stuart Broad has been ruled out of the upcoming two-match Twenty20 series against India with a bruised left heel," an ECB statement read. "Broad will return to the UK immediately to begin a period of rest and rehabilitation."
Broad is due to return to India next month, for the final two matches of a one-day international series.
Twenty-two-year-old Harris is currently in India with the England Performance Programme squad in Pune.
Finn was already to be rested for next week's Twenty20s, but it is hoped he will still be available for the ODI series despite the disc strain in his back.
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