If the sum of the first 11 terms of an arithmetical progression equals that of the first 19 terms, then the sum of its first 30 terms is

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S 11 ​ = S 19 ​ ⇒ 2 11 ​ ( 2 a + 10 d ) = 2 19 ​ ( 2 a + 18 d ) ⇒ 11 ( a + 5 d ) = 19 ( a + 9 d ) ⇒ 8 a + 116 d = 0 ⇒ 2 a + 29 d = 0 . S 30 ​ = 2 30 ​ ( 2 a + 29 d ) = 0 .

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