Let be the set of integers {8,14,20,26,32,....350,356,362,368,374} and be the subset of such that no two elements of have a sum of 382 .
Find the maximum number of elements can have .
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The elements of X are the first 6 2 terms in the arithmetic sequence where the n th term is given by a n = 8 + 6 ( n − 1 ) = 2 + 6 n .
Now this sequence is such that, for k < 6 3 ,
a k + a 6 3 − k = ( 2 + 6 k ) + ( 2 + 6 ( 6 3 − k ) ) = 3 8 2 .
Thus we can place 1 element from each of 3 1 pairs that sum to 3 8 2 into Y , but any additional element placed in Y will necessarily yield a pair of elements that sum to 3 8 2 .
Thus Y can have a maximum of 3 1 elements.