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binomials not successive terms of an arithmetic progression

Source: Polish MO Finals 1974 p5

August 23, 2024
binomial coefficientsalgebraarithmetic sequence

Problem Statement

Prove that for any natural numbers n,rn,r with r+3≤nr + 3 \le n the binomial coefficients (nr)n \choose r, (nr+1)n \choose r+1, (nr+2)n \choose r+2 , (nr+3)n \choose r+3 cannot be successive terms of an arithmetic progression.
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