已知数列递推公式求通项公式的几种方法求数列通项公式的方法一、公式法n例1 已知数列满足，，求数列的通项公式。
{}aa,2{}aaa,，，232n1n，1nn aaaaa33n，1nnn，1nn，1n解:两边除以，得,，，则,,，故数列{}是2aa,，，232，1nnn，1nn，1nn2222222aa3231n,,1以为首项，以为公差的等差数列，由等差数列的通项公式，得，,，,1(1)nn12222231n所以数列的通项公式为an,,。
()2{}ann22aa3nnn，1评注:本题解题的关键是把递推关系式转化为,,，说明数列aa,，，232，1nn，1nn222aa3nn{}是等差数列，再直接利用等差数列的通项公式求出，进而求出数列,，,1(1)nnn222的通项公式。
{}an二、累加法例2 已知数列满足，求数列的通项公式。
{}aaana,，，,211，{}annn，11n 解:由得则 aan,，，21aan,,，21nn，1nn，1aaaaaaaaaa,,，,，，,，,，()()()()？nnnnn,,,11232211,,，，,，，，，，，，，，[2(1)1][2(2)1](221)(211)1nn？,,，,，，，，,，2[(1)(2)21](1)1nnn？(1)nn,,，,，2(1)1n2,,，，(1)(1)1nn2,n2{}a所以数列的通项公式为an,。
nnaan,，，21aan,,，21评注:本题解题的关键是把递推关系式转化为，进而求nn，1nn，1()()()()aaaaaaaaa,，,，，,，,，？{}a出，即得数列的通项公式。
nnnn,,,11232211nadministrative leadership of the criticism, resulting in serious consequences, the pursuit of leadership, along with 500-1000 punishment. (5) significant near miss should be attempted as the case of responsible for the accident and construction team injuries accident penalty provisions, mutatis mutandis. Eight, should perform in the construction standards and specifications, serial number a 1 GB3323-2005 steel fusion welded butt joints, welding engineering-Ray lighting and quality rating of 2 GB11345-89 steel welds manual methods of ultrasonic inspection and testing results for grade 3 GB50236-2002 industrial pipe welding engineering code for construction and acceptance of field equipment 4 HGJ222-92 technical specification for welding of aluminium and its alloys 5 low temperature steel welding procedure 6 SH3525-2004 petrochemical JB/ T4708-2000 of welding procedure qualification forsteel pressure vessels 7 JB/4709-2000 8 JB4730-2005 pressure vessel welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded tank code for construction and acceptance of 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98compressor, fan and pump installation engineering construction and acceptance specification for lifting 6n例3 已知数列满足，求数列的通项公式。
{}a{}aaaa,，，，,2313，nn，11nnnn解:由得则 aa,，，，231aa,,，，231，1，1nnnnaaaaaaaaaa,,，,，，,，,，()()()()？nnnnn,,,11232211nn,,1221,，，，，，，，，，，，，，(231)(231)(231)(231)3？nn,,1221,，，，，，,，2(3333)(1)3？nn,1 3(13),,，,，2(1)3n13,n,,，,，3313nn,，,31nn所以 an,，,31.nnn评注:本题解题的关键是把递推关系式转化为，aa,，，，231aa,,，，231，1，1nnnn进而求出，即得数列的通aaaaaaaaaa,,，,，，,，,，()()()()？{}annnnn,,,11232211n项公式。
n例4 已知数列满足，求数列的通项公式。
{}a{}aaaa,，，，,32313，nn，11nnaa21n，1nnn，1,，，3解:两边除以，得， aa,，，，3231，1nnn，，11nn3333aa21nn，1,,，则，故 nnn，，113333aaaaaaaaaannnnnnn,,,,,11223211,,，,，,，，,，()()()()？nnnnn,,,2232133333333aann,,11212121213,，，，，，，，，，()()()()？ nnn,,1223333333332(1)11111n,,，，，，，，，()1？nnnn,,1223333331n,1,(13)nann,2(1)211n3因此， ,，，,，,1nn,，33133223211nnan,，，，，,33.则 n322administrative leadership of the criticism, resulting in serious consequences, the pursuit of leadership, along with 500-1000 punishment.(5) significant near miss should be attempted as the case of responsible for the accident and construction team injuries accident penalty provisions, mutatis mutandis. Eight, should perform in the construction standards and specifications, serial number a 1 GB3323-2005 steel fusion welded butt joints, welding engineering-Ray lighting and quality rating of 2 GB11345-89 steel welds manual methods of ultrasonic inspection and testing results for grade 3 GB50236-2002 industrial pipe welding engineering code for construction and acceptance of field equipment 4 HGJ222-92 technical specification for welding of aluminium and itsalloys 5 low temperature steel welding procedure 6 SH3525-2004 petrochemical JB/ T4708-2000 of welding procedure qualification forsteel pressure vessels 7 JB/4709-2000 8 JB4730-2005 pressure vessel welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded tank code for construction and acceptance of 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98compressor, fan and pump installation engineering construction and acceptance specification for lifting 6aa21nnn，1评注:本题解题的关键是把递推关系式转化为，,,，aa,，，，3231，1nnn，，11nn3333aaaaaaaaaa,,nnnnnn,,,,,11223211n进而求出，即得数列()()()(),，,，,，，,，？,,nnnnnn,,,,,1122321n3333333333,,的通项公式，最后再求数列的通项公式。