If the A.M. between p th and q th terms of an A.P. is equal to the A.M. between r th and s th terms of the same A.P., then p + q is equal to

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Given: 2 a p ​ + a q ​ ​ = 2 a r ​ + a s ​ ​ ​ ⇒ a + ( p − 1 ) d + a + ( q − 1 ) d = a + ( r − 1 ) d + a + ( s − 1 ) d ⇒ 2 a + ( p + q ) d − 2 d = 2 a + ( r + s ) d − 2 d ⇒ ( p + q ) d = ( r + s ) d ⇒ p + q = r + s . ​

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