{"id":1576,"date":"2024-05-09T11:18:43","date_gmt":"2024-05-09T03:18:43","guid":{"rendered":"http:\/\/jiaohuweike.online\/?p=1576"},"modified":"2024-05-09T11:18:43","modified_gmt":"2024-05-09T03:18:43","slug":"liexiangs","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=1576","title":{"rendered":"\u6570\u5217\u901a\u9879\u53ca\u6027\u8d28"},"content":{"rendered":"\n
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\u6c42\u901a\u9879\u516c\u5f0f<\/h4><\/div>\n\n\n\n
1.(1)\u5df2\u77e5\u6570\u5217\\left \\{ a_n \\right \\} \u6ee1\u8db3S_n=n^2+2n,\u6c42\u901a\u9879\u516c\u5f0fa_n;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u5df2\u77e5\u6570\u5217\\left \\{ a_n \\right \\} \u6ee1\u8db3S_n=2^n+2,\u6c42\u901a\u9879\u516c\u5f0fa_n;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(3)\u5df2\u77e5\u6570\u5217\\left \\{ a_n \\right \\} \u6ee1\u8db3S_n=2^n-1,\u6c42\u901a\u9879\u516c\u5f0fa_n;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(4)\u5df2\u77e5\u6570\u5217\\left \\{ a_n \\right \\} \u6ee1\u8db3a_{1}+3 a_{2}+5 a_{3}+\\cdots+(2 n-1) a_{n}=(n+1)^{2}, \u6c42\u901a\u9879\u516c\u5f0f a_{n}.~~~~~~<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
\u7b54\u6848:(1)a_n=2n+1;(2)\na_n=\\left\\{\\begin{matrix}\n4,n=1& \\\\\n   2^{n-1},n\\ge 2 &\n\\end{matrix}\\right.(3)a_n=   2^{n-1};(4)a_n=\\left\\{\\begin{matrix}\n 4,n=1 & \\\\\n\\displaystyle \\frac{2n+1}{2n-1},n\\ge 2 &\n\\end{matrix}\\right.~~\\\\\u63d0\u793a:\u8ba8\u8bban=1\u548cn\\ge 2\u4e24\u79cd\u60c5\u51b5,\u5373\na_n=\\left\\{\\begin{matrix}\n a_1,n=1 & \\\\\n  S_n-S_{n-1},n\\ge 2&\n\\end{matrix}\\right..~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
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2.\u5df2\u77e5\u6570\u5217\\left \\{ a_n \\right \\} \u6ee1\u8db3a_1=1,a_na_{n+1}=2^n,\u6c42a_n.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
\u7b54\u6848\uff1a2.a_n=\\left\\{\\begin{matrix}\n 2^{\\frac{n-1}{2} },n\u4e3a\u5947\u6570  \\\\\n 2^{\\frac{n}{2} } \uff0cn\u4e3a\u5076\u6570\n\\end{matrix}\\right.;~~\u63d0\u793a\uff1aa_na_{n+1}=2^n\uff0ca_{n-1}a_{n}=2^{n-1}(n\\ge2),\u4e24\u5f0f\u6bd4,~~~~\\\\\\frac{a_{n+1}}{a_{n-1}}=2,\\therefore \u5947\u6570\u9879\u4e3aa_1=1,a_3=2,a_5=2^2,\\dots \u5076\u6570\u9879\u4e3aa_2=2,a_4=2^2,a_6=2^3,\\dots~~~<\/pre><\/div>\n\n\n\n
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3.\u5df2\u77e5\u6570\u5217\\left \\{ a_n \\right \\} \u6ee1\u8db3a_1=2,(n-2)S_{n+1}+2a_{n+1}=nS_n,\u6c42a_n.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
\u7b54\u6848\uff1aa_n=2n;\u63d0\u793a\uff1a\u8f6c\u5316a_{n+1}=S_{n+1}-S_{n},\\Rightarrow\\frac{S{n+1}}{S_n}=\\frac{n+2}{n},\u7d2f\u4e58~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u5f97S_n=n(n+1),\\therefore a_n=2n.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
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4.(1)\u5df2\u77e5\u6570\u5217\\left \\{ a_n \\right \\} \u6ee1\u8db3a_1=4,S_{n}=a_{n+1},\u6c42\u901a\u9879\u516c\u5f0fa_n;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(2)\u5df2\u77e5\u6570\u5217\\left \\{ a_n \\right \\} \u6ee1\u8db3\n6S_n=(3n+2)a_n+2,\u6c42\u901a\u9879\u516c\u5f0fa_n.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
(1)\u7b54\u6848\uff1aa_n=\\left\\{\\begin{matrix}\n  4,n=1 \\\\\n 2^{n},n\\ge 2 \n\\end{matrix}\\right.;\u63d0\u793a\uff1an\\ge2\u65f6\uff0cS_n-S_{n-1}=a_n,\\Rightarrow \\frac{a_{n+1}}{a_n}=2(n\\ge2),~~~~~~~\\\\\u4f46\u662f\\frac{a_2}{a_1}=1\uff0c\\therefore \u9996\u9879\u4e0d\u6ee1\u8db3\u7b49\u6bd4\u6570\u5217.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\(2)\u7b54\u6848:a_n=3n-1.\u63d0\u793a\uff1an\\ge2\u65f6\uff0cS_n-S_{n-1}=a_n.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
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5.\u6570\u5217\\left \\{ a_n \\right \\} \u4e3a\u6b63\u9879\u6570\u5217\uff0ca_1=1,S_n+S_{n+1}=a_{n+1}^2.\u6c42a_n.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
\u7b54\u6848\uff1aa_n=n;\u63d0\u793a:n\\ge 2\uff0cS_{n-1}+S_{n}=a_{n}^2\uff0c\u4e24\u5f0f\u76f8\u51cf\u5f97a_n-a_{n-1}=1.~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
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6.\u6b63\u9879\u6570\u5217  \\left\\{a_{n}\\right\\}  \u6ee1\u8db3\na_{1}=2,6 S_{n}=\\left(a_{n}+2\\right)\\left(a_{n}+1\\right) \\text ,\u6c42\u6570\u5217  \\left\\{a_{n}\\right\\}  \u7684\u901a\u9879\u516c\u5f0f.~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
\u7b54\u6848\uff1aa_n=3n-1\uff1b\u5229\u7528S_{n}-S_{n-1}=a_n\u8f6c\u5316.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
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7.\u5df2\u77e5\u7b49\u5dee\u6570\u5217  \\left\\{a_{n}\\right\\} , \u6ee1\u8db3  \\frac{a_{n}+a_{n+1}}{4}=n+1 ,\u6c42  \\left\\{a_{n}\\right\\}  \u7684\u901a\u9879\u516c\u5f0f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
\u7b54\u6848\uff1aa_n=2n+1\uff1b\u63d0\u793a\uff1a\\therefore  \\left\\{a_{n}\\right\\} \u4e3a\u7b49\u5dee\u6570\u5217\uff0c\u4ee4n=1,n=2,\u89e3\u5f97a_1=3,d=2.~~~~<\/pre><\/div>\n\n\n\n
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8.\u5df2\u77e5\u7b49\u5dee\u6570\u5217  \\left\\{a_{n}\\right\\}  \u7684\u9996\u9879\u4e3a 1 , \u524d  n  \u9879\u548c\u4e3a  S_{n} . \u8bb0  b_{n}=\\frac{S_{n}+n}{n(n+1)} ,\u6570\u5217  \\left\\{b_{n}\\right\\}  \u662f\u7b49\u5dee\u6570\u5217,\\\\\n\u6c42\u6570\u5217  \\left\\{a_{n}\\right\\}  \u7684\u901a\u9879\u516c\u5f0f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
\u7b54\u6848\uff1aa_n=2n-1;\u63d0\u793a\uff1a\u8bbe\u6570\u5217  \\left\\{a_{n}\\right\\} \u516c\u5dee\u4e3ad,\u53c8\u56e0\u6570\u5217  \\left\\{b_{n}\\right\\}  \u662f\u7b49\u5dee\u6570\u5217\uff0c\u6240\u4ee5~~~~~~~~\\\\b_1+b_3=2b_2,\\Rightarrow d =2,a_n=2n-1.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
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9.\u6b63\u9879\u6570\u5217  \\left\\{a_{n}\\right\\}  \u6ee1\u8db3  a_{1}=1, \\frac{a_{n}}{a_{n+1}}=a_{n}+2\\left(n \\in N^{*}\\right) ,\u8bc1\u660e: \u6570\u5217  \\left\\{\\frac{1}{a_{n}}+1\\right\\}  \u662f\u7b49\u6bd4~~~~~\\\\\u6570\u5217\uff0c\u5e76\u6c42\u901a\u9879\u516c\u5f0fa_n.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
\u7b54\u6848\uff1aa_n=\\frac{1}{2^n-1}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
\n<\/details>\n\n\n\n
10.(1) \u5728\u6570\u5217  \\left\\{a_{n}\\right\\}  \u4e2d,  a_{1}=2, a_{n+1}=a_{n}+\\ln \\left(1+\\frac{1}{n}\\right) , \u6c42\u6570\u5217  \\left\\{a_{n}\\right\\}  \u7684\u901a\u9879\u516c\u5f0f  a_{n}.~~~~~~~\n\n\n\n<\/pre><\/div>\n\n\n\n
(2) \u5728\u6570\u5217  \\left\\{a_{n}\\right\\}  \u4e2d,a_{n+1}=\\frac{n}{n+2} a_{n}\\left(n \\in \\mathbf{N}^{*}\\right) , \u4e14  a_{1}=4 , \u6c42\u6570\u5217  \\left\\{a_{n}\\right\\}  \u7684\u901a\u9879\u516c\u5f0f  a_{n}.~~~~~~~<\/pre><\/div>\n\n\n\n
(3) \u5728\u6570\u5217  \\left\\{a_{n}\\right\\}  \u4e2d,\u82e5  a_{1}=1, a_{n+1}=2 a_{n}+3 , \u6c42\u6570\u5217  \\left\\{a_{n}\\right\\}  \u7684\u901a\u9879\u516c\u5f0f  a_{n}.~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
(4)\u5728\u6570\u5217  \\left\\{a_{n}\\right\\}  \u4e2d,\u82e5  a_{1}=1, a_{n+1}=2 a_{n}+2^{n+1} , \u6c42\u6570\u5217  \\left\\{a_{n}\\right\\}  \u7684\u901a\u9879\u516c\u5f0f  a_{n}.~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
(5)\u82e5\u6570\u5217  \\left\\{a_{n}\\right\\}  \u6ee1\u8db3  a_{1}=1, a_{n+1}=\\frac{2 a_{n}}{a_{n}+2} , \u6c42\u6570\u5217  \\left\\{a_{n}\\right\\}  \u7684\u901a\u9879\u516c\u5f0f  a_{n}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n
\u7b54\u6848<\/summary>\n \n\n\n\n
\u7b54\u6848:(1) a_{n}=2+\\ln n ;\u63d0\u793a:\u7d2f\u52a0\u6cd5;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(2)\u7b54\u6848:a_{n}=\\frac{8}{n(n+1)} ;\u63d0\u793a:\u7d2f\u4e58\u6cd5;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(3)\u7b54\u6848 :a_n= 2^{n+ 1}-3 ; \u63d0\u793a: \u6784\u9020\u6cd5.\u5f62\u5982  a_{n+1}=A a_{n}+B , \u4e24\u8fb9\u540c\u52a0X,\\Rightarrow ~~~~~~~~~~~~~~~~~~\\\\a_{n+1}+X=Aa_n+B+X=A\\left ( a_n+\\frac{B+X}{A}  \\right )   ,a_{n+1}+3=2\\left(a_{n}+3\\right)\\Rightarrow a_{n}=2^{n+1}-3 .\\\\\n(4)\u7b54\u6848:a_n=(n-\\frac{1}{2})2^{n};\u63d0\u793a: \u4e24\u8fb9\u540c\u9664\u4ee5  2^{n+1}, \\Rightarrow \\frac{a_{n+1}}{2^{n+1}}=\\frac{a_{n}}{2^{n}}+1 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(5)\u7b54\u6848:a_n=  \\frac{2}{n+1} , \u63d0\u793a: \u53d6\u5012\u6570\uff0c \\frac{1}{a_{n+1}}=\\frac{1}{a_{n}}+\\frac{1}{2}, \\frac{1}{a_{n}}=\\frac{n}{2}+\\frac{1}{2} , \u6240\u4ee5  a_{n}=\\frac{2}{n+1} .~~~\n<\/pre><\/div>\n<\/details>\n\n\n\n