给定100个实数a1,a2,……,a100,满足a1-4a2+3a3≥0,a2-4a3+3a4≥0,a3-4a4+3a5≥0,……,a100-4a1+3a2≥0,求证：这一百个数全相等

题目

给定100个实数a1,a2,……,a100,满足a1-4a2+3a3≥0,a2-4a3+3a4≥0,a3-4a4+3a5≥0,……,a100-4a1+3a2≥0,求证：这一百个数全相等

答案

(a1-a2)>=3(a2-a3)
(a2-a3)>=3(a3-a4)
.
(a99-a100)>=3(a100-a1)
(a100-a1)>=3(a1-a2)
因此逐项代入即得：(a1-a2)>=3^99*(a1-a2),得：a1-a2