diff --git a/工具/文本文件/批量题目分类号记录.txt b/工具/文本文件/批量题目分类号记录.txt
index 561878d6..c65a1ff4 100644
--- a/工具/文本文件/批量题目分类号记录.txt
+++ b/工具/文本文件/批量题目分类号记录.txt
@@ -18,7 +18,7 @@ problems_dict = {
 "新教材必修第一册课堂练习": "009426:009538",
 "新教材必修第一册习题": "010017:010202",
 "新教材必修第一册复习题": "1:94",
-"空中课堂数列例题与习题": "1:1",
+"空中课堂数列例题与习题": "18258:18314",
 "新教材数列课堂练习": "9875:9904",
 "新教材数列习题": "10741:10788",
 "新教材数列复习题": "306:325",
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index b284bc2b..027f42bb 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -471353,6 +471353,1146 @@
         "space": "4em",
         "unrelated": []
     },
+    "018258": {
+        "id": "018258",
+        "content": "已知等差数列$-5,-9,-13, \\cdots$.\\\\\n(1) 求该等差数列的第$20$项;\\\\\n(2) $-401$ 是不是该等差数列的项? 如果是, 指明是第几项; 如果不是, 请说明理由.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018259": {
+        "id": "018259",
+        "content": "假设体育场一角看台的座位从第$2$排起每一排都比前一排多相等数目的座位. 若第$3$排有$10$个座位, 第$9$排有$28$个座位, 则第$12$排有多少个座位?",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018260": {
+        "id": "018260",
+        "content": "已知$a_n=p n+q$是数列$\\{a_n\\}$的通项公式, 其中$p$和$q$均为常数. 试判断数列$\\{a_n\\}$是否为等差数列, 并证明你的结论.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018261": {
+        "id": "018261",
+        "content": "若在$7$和 $21$ 中插入$3$个数, 使这$5$个数成等差数列, 求这$3$个数.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018262": {
+        "id": "018262",
+        "content": "在等差数列$\\{a_n\\}$中, 若$a_n=m$, $a_m=n$, 且$n \\neq m$, 求$a_{m+n}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018263": {
+        "id": "018263",
+        "content": "设数列$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_1=50$, $a_8=15$, 求$S_8$;\\\\\n(2) 已知$a_1=0.7$, $a_2=1.5$, 求$S_7$;\\\\\n(3) 已知$a_4=7$, 求$S_7$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018264": {
+        "id": "018264",
+        "content": "已知等差数列$\\{a_n\\}$的前$10$项和$S_{10}=310$, 前$20$项和$S_{20}=1220$, 由此可以确定数列$\\{a_n\\}$前$30$项和$S_{30}$吗?",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018265": {
+        "id": "018265",
+        "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=n^2+2 n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求证: 数列$\\{a_n\\}$是等差数列.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018266": {
+        "id": "018266",
+        "content": "已知两个等差数列$2,6,10, \\cdots, 190$及$2,8,14, \\cdots, 200$, 将这两个等差数列的公共项按从小到大的顺序组成一个新数列. 求这个新数列的所有项之和.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018267": {
+        "id": "018267",
+        "content": "设数列$\\{a_n\\}$为等比数列.\\\\\n(1) 已知$a_1=3$, 公比$q=-2$, 求$a_6$;\\\\\n(2) 已知$a_3=20$, $a_6=160$, 求$a_n$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018268": {
+        "id": "018268",
+        "content": "某种放射性物质不断衰变为其他物质, 设每经过一年剩余的这种放射性物质是年初的$84 \\%$. 这种放射性物质的半衰期约为多少? (结果精确到$1$年)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018269": {
+        "id": "018269",
+        "content": "已知$a, b, c$成等差数列, 其公差为$d$. 试证明$3^a, 3^b, 3^c$成等比数列, 并写出其公比.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018270": {
+        "id": "018270",
+        "content": "已知正实数$a, b, c$成等比数列, 其公比为$q$. 试证明$\\lg a, \\lg b, \\lg c$成等差数列, 并写出其公差.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018271": {
+        "id": "018271",
+        "content": "用$10000$元购买某个理财产品一年.\\\\\n(1) 若以月利率$0.400 \\%$的复利计息, $12$个月能获得多少利息? (结果精确到$1$元)\\\\\n(2) 若以季度复利计息, 存$4$个季度, 则当每季度利率为多少时, 按季结息的利息不少于按月结息的利息? (结果精确到$0.001 \\%$)\n(复利是指把前一期的利息和本金加在一起算作本金, 再计算下一期的利息)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018272": {
+        "id": "018272",
+        "content": "已知项数无限的数列$\\{a_n\\}$为等比数列, 公比为$q$.\\\\\n(1) 将数列$\\{a_n\\}$中的前$k$项去掉, 剩余项组成一个新数列, 这个新数列是等比数列吗? 如果是, 它的首项与公比分别是多少?\\\\\n(2) 取出数列$\\{a_n\\}$中的所有奇数项, 组成一个新数列, 这个新数列是等比数列吗? 如果是, 它的首项与公比分别是多少?\\\\\n(3) 在数列$\\{a_n\\}$中, 每隔$10$项取出一项, 组成一个新数列, 这个新数列是等比数列吗? 如果是, 它的公比是多少? 你能根据本小题得到的结论作出关于等比数列$\\{a_n\\}$的一个更一般的猜想吗? 尝试证明你的猜想.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018273": {
+        "id": "018273",
+        "content": "设数列$\\{a_n\\}$为等比数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_1=-4$, 公比$q=-\\dfrac{1}{2}$, 求$S_{10}$;\\\\\n(2)已知$a_1=27$, $a_n=\\dfrac{1}{243}$, 公比$q=-\\dfrac{1}{3}$, 求$S_n$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018274": {
+        "id": "018274",
+        "content": "在等比数列$\\{a_n\\}$中, 其前$n$项和为$S_n$. 已知$S_3=\\dfrac{7}{2}$, $S_6=\\dfrac{63}{2}$, 求$a_n$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018275": {
+        "id": "018275",
+        "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=3^n+a$($a$是实数). 当常数$a$满足什么条件时, 数列$\\{a_n\\}$是等比数列?",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018276": {
+        "id": "018276",
+        "content": "设数列$\\{a_n\\}$, 其前$n$项和为$S_n$, 且$S_n=2 a_n+1$, 求$S_n$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018277": {
+        "id": "018277",
+        "content": "设数列$\\{a_n\\}$, 其前$n$项和为$S_n$, 且$a_n+S_n=2$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 若数列$\\{b_n\\}$满足$b_n=a_n+2 n$, 求数列$\\{b_n\\}$的前$n$项和$T_n$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018278": {
+        "id": "018278",
+        "content": "某人今年初向银行申请贷款$20$万元, 月利率为$3.375 \\%$, 按复利计算, 每月等额还贷一次, 并从贷款后的次月初开始还贷. 如果$10$年还清, 那么每月应还贷多少元? (结果精确到 0.01 元)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018279": {
+        "id": "018279",
+        "content": "化下列循环小数为分数:\\\\\n(1) $0 . \\dot{2} \\dot{9}$;\\\\\n(2) $0.4 \\dot{3} \\dot{1}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018280": {
+        "id": "018280",
+        "content": "如图, 正方形$ABCD$的边长等于 1, 连接这个正方形各边的中点得到一个小正方形$A_1B_1C_1D_1$; 又连接正方形$A_1B_1C_1D_1$各边的中点得到一个更小的正方形$A_2B_2C_2D_2$; 如此无限继续下去. 求所直这些正方形的周长的和与面积的和.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 0.6]\n\\draw (0, 0) rectangle (4, 4);\n\\draw (0, 2) -- (2, 4) -- (4, 2) -- (2, 0) -- cycle;\n\\draw (1, 3) -- (3, 3) -- (3, 1) -- (1, 1) -- cycle;\n\\draw (2, 3) -- (3, 2) -- (2, 1) -- (1, 2) -- cycle;\n\\draw (0, 4) node [above left] {$A$} (4, 4) node [above right] {$B$} (4, 0) node [below right] {$C$} (0, 0) node [below left] {$D$};\n\\draw (2, 4) node [above] {$A_1$} (4, 2) node [right] {$B_1$} (2, 0) node [below] {$C_1$} (0, 2) node [left] {$D_1$};\n\\draw (3, 3) node [above right] {$A_2$} (3, 1) node [below right] {$B_2$} (1, 1) node [below left] {$C_2$} (1, 3) node [above left] {$D_2$};\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018281": {
+        "id": "018281",
+        "content": "计算$\\displaystyle\\sum_{i=1}^{+\\infty}(\\dfrac{1}{4})^{i-1}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018282": {
+        "id": "018282",
+        "content": "化下列循环小数为分数:\\\\\n(1) $0 . \\dot{1} \\dot{4}$;\\\\\n(2) $1.2 \\dot{3}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018283": {
+        "id": "018283",
+        "content": "已知$\\{a_n\\}$是等比数列, $a_2=2$, $a_5=\\dfrac{1}{4}$, 求$\\displaystyle\\sum_{i=1}^{+\\infty} a_i a_{i+1}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018284": {
+        "id": "018284",
+        "content": "正六边形$ABCDEF$的边长为$1$, 连接这个正六边形各边的中点得到一个小正六边形$A_1B_1C_1D_1E_1F_1$. 又连接正六边形$A_1B_1C_1D_1E_1F_1$各边的中点得到一个更小的正六边形$A_2B_2C_2D_2E_2F_2$, 如此无限继续下去. 求所有这些正六边形的周长之和.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018285": {
+        "id": "018285",
+        "content": "查阅相关史料, 了解在极限思想的发展史上数学家作出的一些贡献, 以及了解古今中外利用极限思想解决的一些经典问题, 谈谈自己的感悟体会, 写一篇$500$字左右的小作文.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018286": {
+        "id": "018286",
+        "content": "已知数列$\\{a_n\\}$的通项公式, 写出这些数列的前$5$项:\\\\\n(1) $a_n=\\dfrac{n-2}{n+1}$;\\\\\n(2) $a_n=1+(-\\dfrac{1}{2})^n$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018287": {
+        "id": "018287",
+        "content": "给出数列$\\{a_n\\}$的下述通项公式, 判断这些数列是否为单调数列, 请说明理由.\\\\\n(1) $a_n=1+(\\dfrac{1}{2})^n$;\\\\\n(2) $a_n=n-\\dfrac{1}{n}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018288": {
+        "id": "018288",
+        "content": "已知数列$\\{a_n\\}$的通项公式是$a_n=(n+1)(\\dfrac{9}{10})^{n-1}$. 试问: 该数列是否有最大项? 若有, 指出第几项最大; 若没有, 试说明理由.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018289": {
+        "id": "018289",
+        "content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 判断``数列$\\{a_n\\}$各项均为正数''是``数列$\\{S_n\\}$是严格增数列''的什么条件? 说明理由.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018290": {
+        "id": "018290",
+        "content": "设等比数列$\\{a_n\\}$中首项$a_1=1$, 公比为$q(q \\neq 0)$, 试根据公比$q$的不同取值, 讨论数列$\\{a_n\\}$是否有最大项, 说明理由.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018291": {
+        "id": "018291",
+        "content": "在平面上画$n$条直线, 假设其中任意$2$条直线都相交, 且任意$3$条直线都不共点. 设这$n$条直线将平面分成了$a_n$个部分.\\\\\n(1) 写出数列$\\{a_n\\}$的一个递推公式;\\\\\n(2) 写出数列$\\{a_n\\}$的一个通项公式.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018292": {
+        "id": "018292",
+        "content": "已知数列$\\{a_n\\}$满足$\\begin{cases}a_n=2 a_{n-1}+1(n \\geq 2), \\\\ a_1=1 .\\end{cases}$\\\\\n(1) 求证: 数列$\\{a_n+1\\}$为等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018293": {
+        "id": "018293",
+        "content": "已知数列$\\{a_n\\}$各项均为正数, 满足$a_n^2=a_{n-1}^2+1$($n \\geq 2$)且$a_1=1$, 求数列$\\{a_n\\}$的通项公式.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018294": {
+        "id": "018294",
+        "content": "设数列$\\{a_n\\}$满足$\\begin{cases}a_n=a_{n-1}-a_{n-2}(n \\geq 3), \\\\ a_1=a, \\\\ a_2=b,\\end{cases}$其中常数$a$、$b$为实数.\\\\\n(1) 求证: $a_{n+3}=-a_n$;\\\\\n(2) 证明$a_{n+6}=a_n$, 并求数列的前$2022$项和$S_{2022}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018295": {
+        "id": "018295",
+        "content": "用数学归纳法证明: $1+3+5+\\cdots+(2 n-1)=n^2$($n$为正整数).",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018296": {
+        "id": "018296",
+        "content": "用数学归纳法证明: $1^3+2^3+3^3+\\cdots+n^3=[\\dfrac{n(n+1)}{2}]^2$($n$为正整数).",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018297": {
+        "id": "018297",
+        "content": "若$P(n)$表示等式$1-\\dfrac{1}{2}+\\dfrac{1}{3}-\\dfrac{1}{4}+\\cdots+\\dfrac{1}{n-1}-\\dfrac{1}{n}=2(\\dfrac{1}{n+2}+\\dfrac{1}{n+4}+\\cdots+\\dfrac{1}{2 n})(n$为正偶数), 则$P(2)$表示的等式为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018298": {
+        "id": "018298",
+        "content": "设$f(n)=1+\\dfrac{1}{2}+\\dfrac{1}{3}+\\cdots+\\dfrac{1}{n}$($n$为正整数), 那么$f(2^{n+1})-f(2^n)=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018299": {
+        "id": "018299",
+        "content": "用数学归纳法证明: $1^2-2^2+3^2-4^2+\\cdots+(-1)^{n-1} \\cdot n^2=(-1)^{n-1} \\cdot \\dfrac{n(n+1)}{2}(n$为正整数).",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018300": {
+        "id": "018300",
+        "content": "已知数列$\\{a_n\\}$满足$\\begin{cases}a_{n+1}=a_n+\\dfrac{n}{a_n}, \\\\ a_1=1 .\\end{cases}$尝试通过计算数列$\\{a_n\\}$的前四项, 猜想数列$\\{a_n\\}$的通项公式, 并用数学归纳法加以证明.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018301": {
+        "id": "018301",
+        "content": "是否存在常数$a$、$b$、$c$, 使等式$1^2+2^2+3^2+\\cdots+n^2=a n^3+b n^2+c n$对任意正整数$n$都成立?",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018302": {
+        "id": "018302",
+        "content": "设数列$\\{a_n\\}$, 其前$n$项和为$S_n$, 且$S_n+a_n=n$. 计算$a_1, a_2, a_3$; 根据计算的结果, 猜想数列$\\{a_n\\}$的通项公式, 并用数学归纳法加以证明.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018303": {
+        "id": "018303",
+        "content": "观察下列数字:\n\\begin{center}\n\\begin{tabular}{ccccccccc}\n1 & & & & & & \\\\\n2 & 3 & 4 & & & & \\\\\n3 & 4 & 5 & 6 & 7 & & \\\\\n4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n$\\cdots$ &$\\cdots$&$\\cdots$&$\\cdots$&$\\cdots$&$\\cdots$&$\\cdots$&$\\cdots$&$\\cdots$\n\\end{tabular}\n\\end{center}\n猜想第$n$($n$为正整数) 行的所有数之和$S_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018304": {
+        "id": "018304",
+        "content": "依次计算$(1-\\dfrac{1}{2})$, $(1-\\dfrac{1}{2})(1-\\dfrac{1}{3})$, $(1-\\dfrac{1}{2})(1-\\dfrac{1}{3})(1-\\dfrac{1}{4})$的值; 根据计算的结果, 猜想$T_n=(1-\\dfrac{1}{2})(1-\\dfrac{1}{3})(1-\\dfrac{1}{4}) \\cdots(1-\\dfrac{1}{n+1})$($n$为正整数)的表达式, 并用数学归纳法加以证明.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018305": {
+        "id": "018305",
+        "content": "已知数列$\\{a_n\\}$满足条件$(n-1) a_{n+1}=(n+1)(a_n-1)$($n$为正整数), 且$a_2=6$, 则$a_1=$\\blank{50}, $a_3=$\\blank{50}, $a_4=$\\blank{50}, 进而猜想$a_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018306": {
+        "id": "018306",
+        "content": "已知数列$\\{a_n\\}$满足$a_1=1$, 且$a_n=2 a_{n-1}+\\dfrac{n+2}{n(n+1)}$($n \\geq 2$).\\\\\n(1) 求$a_2, a_3, a_4$;\\\\\n(2) 猜想数列$\\{a_n\\}$的通项公式, 并用数学归纳法加以证明.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018307": {
+        "id": "018307",
+        "content": "请以四人小组为单位, 构造计算$\\sqrt{3}$的迭代算法的递推公式, 并选取初值$x_1=1$, 列出该迭代序列$\\{x_n\\}$的前$5$项.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018308": {
+        "id": "018308",
+        "content": "请以四人小组为单位, 构造计算$\\sqrt[3]{2}$的迭代算法的递推公式, 并选取初值$x_1=1$, 列出该迭代序列$\\{x_n\\}$的前$5$项.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018309": {
+        "id": "018309",
+        "content": "仿照计算$\\sqrt{2}$的巴比伦算法, 构造计算$\\sqrt{5}$的迭代算法的递推公式, 并选取初值$x_1=1$. 通过计算器操作, 写出该迭代序列$\\{x_n\\}$的前$5$项.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018310": {
+        "id": "018310",
+        "content": "请同学们查阅《九章算术》, 了解中国古代数学算法方面的成就, 如更相减损术、盈亏术、方程术、球积术等.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018311": {
+        "id": "018311",
+        "content": "设数列$\\{a_n\\}$, 其前$n$项和为$S_n$, $a_1=\\dfrac{1}{2}$.\\\\\n(1) 若$\\{a_n\\}$为等比数列, 公比为$q$($0<|q|<1$), $S_3=\\dfrac{7}{8}$, 求$\\displaystyle\\sum_{i=1}^{+\\infty} a_i$;\\\\\n(2) 若$\\{a_n\\}$为等差数列, $a_m=-2$, $a_{m+2}=-3$($m$为正整数), 求$S_m$的值;\\\\\n(3) 若$S_n=2 n^2-n-t$($t \\in \\mathbf{R}$), 试判断数列$\\{a_n\\}$是否为等差数列, 并说明理由;\\\\\n(4) 若$S_n=n-5 a_n+2$, 证明$\\{a_n-1\\}$是等比数列, 并求$\\{a_n\\}$的通项公式.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018312": {
+        "id": "018312",
+        "content": "某生产厂商为提升生产工艺, 研发了新的生产线. 新生产线的使用寿命为$4$年. 该厂商原来每个月能生产产品$1000$个, 产品合格率为$85 \\%$. 从$2021$年$1$月初开始使用新的生产线后, 每个月的产量都在前一个月的基础上提高$5 \\%$, 同时, 当月产品的不合格率与前一个月产品的不合格率之差为$-0.3 \\%$.\\\\\n(1) 求$2021$年$12$月的产量以及不合格品的数量; (结果精确到$1$个)\\\\\n(2) 在生产过程中, 若出现某个月的不合格产品数量超过$300$个, 则生产线会被强制停用. 请分析该生产线是否存在由于上述原因而被强制停用的可能? 并说明理由.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018313": {
+        "id": "018313",
+        "content": "据测算, 如果不加处理, 每吨工业废弃垃圾将占地$1$平方米. 环保部门每回收或处理$1$吨废旧物资, 相当于消灭$4$吨工业废弃垃圾. 如果某环保部门从去年开始回收处理废旧物资, 第$1$年共回收处理了$10^4$吨废旧物资, 且以后每年的回收量比上一年递增$20 \\%$.\\\\\n(1) 第$7$年能回收多少吨废旧物资? (结果用科学记数法表示, 保留一位小数)\\\\\n(2) 前$7$年共节约土地多少平方米? (结果用科学记数法表示, 保留一位小数)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018314": {
+        "id": "018314",
+        "content": "1934 年, 东印度(今孟加拉国)学者森德拉姆(Sundaram)发现了``正方形筛子'':\n\\begin{center}\n\\begin{tabular}{cccccc}\n4 & 7 & 10 & 13 & 16 &$\\cdots$\\\\ \n7 & 12 & 17 & 22 & 27 &$\\cdots$\\\\ \n10 & 17 & 24 & 31 & 38 &$\\cdots$\\\\ \n13 & 22 & 31 & 40 & 49 &$\\cdots$\\\\ \n16 & 27 & 38 & 49 & 60 &$\\cdots$\\\\ \n$\\cdots$ & $\\cdots$ & $\\cdots$ & $\\cdots$ & $\\cdots$ & $\\cdots$\n\\end{tabular}\n\\end{center}\n(1) ``正方形筛子''中位于第$100$行的第$100$个数是多少?\\\\\n(2) 请你查找相关资料, 尝试与你的学习伙伴说一说这个``正方形筛子''的奥妙之处.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "",
+        "edit": [
+            "20230704\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "020001": {
         "id": "020001",
         "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",