MathLabs

Problem 3

For 2k real numbers a_1, ..., a_k, b_1, ..., b_k, define X_n=sum from i=1 to k of floor(a_i n+b_i), for n=1,2,... . If the sequence X_n is an arithmetic progression, prove that sum from i=1 to k of a_i is an integer. Here floor(r) denotes the greatest integer less than or equal to r.
Step 3 of 5: Use the arithmetic progression
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Xn=X1+(n−1)d,nd=Xn+1−X1,A+B−k<X1≤A+B.X_n=X_1+(n-1)d,\quad nd=X_{n+1}-X_1,\quad A+B-k<X_1\le A+B.
Detailed analysis

Let d be the common difference. Since every term of the sequence is an integer, d, the difference of the second and first terms, is an integer. Writing the next term as nd plus the first term and applying the squeeze at the next index produces a narrow interval.