江西省2023-2024学年度八年级阶段性练习（二）数学.考卷答案

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江西省2023-2024学年度八年级阶段性练习（二）数学.考卷答案试卷答案

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分析由a_n=$\frac{1}{（n+1）（n+2）}$=$\frac{1}{n+1}-\frac{1}{n+2}$，利用“裂项求和”可得S_n，再利用数列的单调性即可得出．

解答解：∵a_n=$\frac{1}{（n+1）（n+2）}$=$\frac{1}{n+1}-\frac{1}{n+2}$，
∴S_n=$（\frac{1}{2}-\frac{1}{3}）$+$（\frac{1}{3}-\frac{1}{4}）$+…+$（\frac{1}{n+1}-\frac{1}{n+2}）$=$\frac{1}{2}-\frac{1}{n+2}$．
∵S_n=$\frac{1}{2}-\frac{1}{n+2}$＜t对任意n∈N^*都成立，
∴$t≥\frac{1}{2}$．
故答案为：$t≥\frac{1}{2}$．

点评本题考查了“裂项求和”、数列的单调性、不等式的性质，考查了推理能力与计算能力，属于中档题．