The first 2n positive integers are arbitrarily divided into two groups of n numbers each. The numbers in the first group are sorted in ascending order: a_{1} < a_{2} < ... < a_{n}; the numbers in the second group are sorted in descending order: b_{1} > b_{2} > ... > b_{n}.

Find, with proof, the value of the sum |a_{1} − b_{1}| + |a_{2} − b_{2}| + ... + |a_{n} − b_{n}|.

The table below is generated at random, in two stages. Firstly, a value of n in the range 2..20 is chosen at random. Then, n numbers in the range 1..2n are chosen at random. These numbers are sorted to form the sequence a_{i}; the remaining numbers are sorted to form the sequence b_{i}. For each i, the value |a_{i} − b_{i}| is calculated, and the sum is given.

Here are some generated random table . Use the tables to formulate a conjecture about each pair, (a_{i}, b_{i}.) Prove your conjecture, and hence find the value of the sum.

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## BMAD 65

The first 2n positive integers are arbitrarily divided into two groups of n numbers each. The numbers in the first group are sorted in ascending order: a

_{1}< a_{2}< ... < a_{n}; the numbers in the second group are sorted in descending order: b_{1}> b_{2}> ... > b_{n}.Find, with proof, the value of the sum |a

_{1}− b_{1}| + |a_{2}− b_{2}| + ... + |a_{n}− b_{n}|.The table below is generated at random, in two stages. Firstly, a value of n in the range 2..20 is chosen at random. Then, n numbers in the range 1..2n are chosen at random. These numbers are sorted to form the sequence a

_{i}; the remaining numbers are sorted to form the sequence b_{i}. For each i, the value |a_{i}− b_{i}| is calculated, and the sum is given.Here are some generated random table . Use the tables to formulate a conjecture about each pair, (a

_{i}, b_{i}.) Prove your conjecture, and hence find the value of the sum.Edited by BMAD## Link to post

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