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添加若干自拟题目

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wangweiye7840 2023-12-25 14:42:41 +08:00
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题库0.3/Problems.json
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@ -615347,6 +615347,266 @@
"space": "",
"unrelated": []
},
"022972": {
"id": "022972",
"content": "在数列 $\\{a_n\\}$ 中, 若 $a_1=3$, 且 $2 a_{n+1}=2 a_n+1$, 则 $a_9=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"022973": {
"id": "022973",
"content": "若 5 与 $x$ 的等比中项为 $5 x$, 则 $x$ 的值为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"022974": {
"id": "022974",
"content": "计算: $3+\\dfrac{3}{2}+\\dfrac{3}{2^2}+\\dfrac{3}{2^3}+\\cdots+\\dfrac{3}{2^{n-1}}=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"022975": {
"id": "022975",
"content": "求和: $1+(1+2)+(1+2+2^2) \\cdots+(1+2+2^2+\\cdots+2^n)=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"022976": {
"id": "022976",
"content": "将正整数 $1,2,3, \\cdots, n, \\cdots$ 按``第 $k$ ($k$ 为正整数 ) 组含 $2^{k-1}$ 个数''的规则分组: $(1),(2,3),(4,5,6,7), \\cdots$, 那么数 $2023$ 在第\\blank{50}组第\\blank{50}个.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"022977": {
"id": "022977",
"content": "已知 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=n^2-4 n+1$, 则 $\\displaystyle\\sum_{i=1}^n|a_i|=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"022978": {
"id": "022978",
"content": "灰白两种颜色的正六边形地面砖按下图的规律拼成若干个图案:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\fill [gray!50] (1,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\foreach \\i in {0,60,...,300}\n{\\draw (\\i:1) --++ (\\i:1);\n\\draw (\\i:1) -- ({\\i+60}:1);\n\\draw (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\draw (0,-4) node {第1个};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\fill [gray!50] (1,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\fill [gray!50] (4,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\foreach \\i in {0,60,...,300}\n{\\draw (\\i:1) --++ (\\i:1);\n\\draw (\\i:1) -- ({\\i+60}:1);\n\\draw (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\foreach \\i in {0,60,...,300}\n{\\draw (3,0) ++ (\\i:1) --++ (\\i:1);\n\\draw (3,0) ++ (\\i:1) --++ ({\\i+120}:1);\n\\draw (3,0) ++ (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\draw (1.5,-4) node {第2个};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\fill [gray!50] (1,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\fill [gray!50] (4,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\fill [gray!50] (7,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) -- cycle;\n\\foreach \\i in {0,60,...,300}\n{\\draw (\\i:1) --++ (\\i:1);\n\\draw (\\i:1) -- ({\\i+60}:1);\n\\draw (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\foreach \\i in {0,60,...,300}\n{\\draw (3,0) ++ (\\i:1) --++ (\\i:1);\n\\draw (3,0) ++ (\\i:1) --++ ({\\i+120}:1);\n\\draw (3,0) ++ (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\foreach \\i in {0,60,...,300}\n{\\draw (6,0) ++ (\\i:1) --++ (\\i:1);\n\\draw (6,0) ++ (\\i:1) --++ ({\\i+120}:1);\n\\draw (6,0) ++ (\\i:2) --++ ({\\i+60}:1) --++ ({\\i+120}:1) --++ ({\\i+180}:1);};\n\\draw (1.5,-4) node {第3个};\n\\end{tikzpicture}\n\\end{center}\n按此规律继续, 第 $n$ 个图案中有白色地面砖块\\blank{50}块.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
},
"022979": {
"id": "022979",
"content": "已知数列 $\\{a_n\\}$ 满足 $a_{n+1}=-a_n^2+2 a_n$, 且 $a_1=0.9$, 令 $b_n=\\lg (1-a_n)$.\\\\\n(1) 求证: 数列 $\\{b_n\\}$ 是等比数列;\\\\\n(2) 求数列 $\\{\\dfrac{1}{b_n}\\}$ 所有项的和.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"022980": {
"id": "022980",
"content": "用数学归纳法证明: 当 $n \\geq 2$ 时, $\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{2 n}>\\dfrac{13}{24}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"022981": {
"id": "022981",
"content": "已知数列 $\\{a_n\\}$, 满足 $a_1=a$($a>0$), $a_n=\\dfrac{2 a_{n-1}}{1+a_{n-1}}$($n \\geq 2$).\\\\\n(1) 求出 $a_2, a_3, a_4$;\\\\\n(2) 由(1)猜想 $a_n$ 的表达式, 并用数学归纳法证明.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"022982": {
"id": "022982",
"content": "已知数列 $\\{a_n\\}$ 中, 前 $n$ 项和 $S_n$ 满足 $S_n=\\dfrac{3}{2}(a_n-1)$, 探索并猜想通项公式 $a_n$ 的表达式, 并用数学归纳法证明.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"022983": {
"id": "022983",
"content": "在数列 $\\{a_n\\}$ 中, 若 $a_1=3$, $a_2=7$, $a_{n+2}=3 a_{n+1}-2 a_n$, 令$b_n=a_{n+1}-a_n$.\\\\\n(1) 求数列$\\{b_n\\}$的通项公式;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"022984": {
"id": "022984",
"content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=\\begin{cases}2 \\times 3^n,& n \\text{为正奇数},\\\\5-4 n,& n \\text{为正偶数},\\end{cases}$ 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20231225\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"030001": {
"id": "030001",
"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",