摘 要: | 题 37 数列 {an}是正项等差数列 ,对任意k∈N ,试证明 :(Ⅰ )log0 .2 ak+1≤ 12 (log0 .2 ak+log0 .2 ak +2 ) ,(Ⅱ ) ak+12ak1≥ (k + 1)a2 -ka1.证 (Ⅰ )所证不等式等价于a2 k +1≥ak +2 ak.设数列 {an}的公差为d ,则a2 k +1-ak+2 ak=(a1+kd) 2 - [a1+ (k + 1)d][a1+ (k - 1)d]=d2 ≥ 0 .当且仅当d =0 ,即ak=ak+1时等号成立 ,∴log0 .2 ak+1≤ 12 (log0 .2 ak+log0 .2 ak+2 ) .(Ⅱ )由 (Ⅰ )得 ak+2ak+1≤ ak+1ak,∴ ak+2ak+1≤ a2a1,ak+1…
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