Question

7．?dāng)?shù)列通項(xiàng)a_n=$\frac{n-\sqrt{97}}{n-\sqrt{98}}$，前30項(xiàng)中最大項(xiàng)和最小項(xiàng)分別是$\frac{10-\sqrt{97}}{10-\sqrt{98}}$；$\frac{9-\sqrt{97}}{9-\sqrt{98}}$．

Answer 1

分析 a_n=$\frac{n-\sqrt{98}+\sqrt{98}-\sqrt{97}}{n-\sqrt{98}}$=1+$\frac{\sqrt{98}-\sqrt{97}}{n-\sqrt{98}}$，當(dāng)n≤9時(shí)，數(shù)列{a_n}單調(diào)遞增；當(dāng)n≥10時(shí)，數(shù)列{a_n}單調(diào)遞減．即可得出．

解答 解：a_n=$\frac{n-\sqrt{97}}{n-\sqrt{98}}$=$\frac{n-\sqrt{98}+\sqrt{98}-\sqrt{97}}{n-\sqrt{98}}$=1+$\frac{\sqrt{98}-\sqrt{97}}{n-\sqrt{98}}$，
當(dāng)n≤9時(shí)，數(shù)列{a_n}單調(diào)遞增，且a₉＜1；當(dāng)n≥10時(shí)，數(shù)列{a_n}單調(diào)遞減，且a₁₀＞1．
∴前30項(xiàng)中最大項(xiàng)和最小項(xiàng)分別是a₁₀=$\frac{10-\sqrt{97}}{10-\sqrt{98}}$，a₉=$\frac{9-\sqrt{97}}{9-\sqrt{98}}$．
故答案分別為：$\frac{10-\sqrt{97}}{10-\sqrt{98}}$；$\frac{9-\sqrt{97}}{9-\sqrt{98}}$．

點(diǎn)評(píng) 本題考查了數(shù)列與函數(shù)的單調(diào)性，考查了分類討論方法、推理能力與計(jì)算能力，屬于中檔題．