Let define a finite arithmetic progression such such that and let there exist a positive integer such that .
Find , the sum to terms of this arithmetic progression.
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S 2 k − 1 = 2 2 k − 1 ( 2 a + ( 2 k − 1 − 1 ) d ) = 2 2 k − 1 ( 2 a + ( 2 k − 2 ) d ) = 2 2 k − 1 2 ( a + ( k − 1 ) d ) = ( 2 k − 1 ) ( a + ( k − 1 ) d ) a k = 0 a + ( k − 1 ) d = 0 S 2 k − 1 = ( 2 k − 1 ) ( 0 ) ∴ S 2 k − 1 = 0