Problem 4

Prove that for any positive integer

k

, there exists an arithmetic sequence

\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots,\frac{a_k}{b_k}

of rational numbers, where

a_i,b_i

are relatively prime positive integers for each

i=1,2,\ldots,k

, such that the positive integers

a_1,b_1,a_2,b_2,\ldots,a_k,b_k

are all distinct.

Step 3 of 5: Separate numerators and denominators

1\le i<j\le k\Longrightarrow a_i=\frac{K^2}{i}+1>\frac{K^2}{j}+1=a_j,\quad b_i=\frac Ki>\frac Kj=b_j

Detailed analysis

If i<j, then a_i=K^2/i+1>K^2/j+1=a_j and b_i=K/i>K/j=b_j. Hence the a_i are pairwise distinct and the b_i are pairwise distinct.