讲课人:等差数列通项公式前n项和公式性质等比数列通项公式前n项和公式性质等差数列•等差数列;若1,2nnaadnnZ且或1,1nnaadnnZ且若等差数列的首项是,公差为d.则有1a111111nmnnmnaandanmdknbaaaadnnmaand性质23 22,{+}{+}npqnmnpqnmmkmkmknnnnnaaabnpqaaaamnpqaaaaaaaaaabaab等差中项:三个数,G,b组成的等差数列,则称G为与b的等差中项2G=若{}是等差数列,则若{}是等差数列,则、、、、构成公差公差kd的等差数列若{}、{}是等差数列则、是等差数列•等差数列的前项和的公式:121122nnnaannSnadpnqn性质*211*212212111nnnnnnnnnnSSndnnSnaaSaSaSSannSnaSnaSnaSnSn偶奇奇偶奇偶奇偶奇偶若项数为,则,若项数为,则,,232SSS,SSS{}mmmmmnn,成等差数列是等差数列,若等差数列,的前n项和为则,nnST1212nnnnTSba•等差数列的求和最值问题:(二次函数的配方法;通项公式求临界项法)若则有最大值,当n=k时取到的最大值k满足001dans001kkaa若则有最大值,当n=k时取到的最大值k满足001dans001kkaa等比数列通项公式及其性质若等比数列的首项,公比为q,则na1a1111,nnmnmnnmnnmaaqaqaaqqaa22232{}npqnmnpqkmmkmkmkaaGabnpqaaaamnpqaaaaaaaaq,G,b成等比数列,则称G为与b的等比中项性质:若是等比数列,则、、、、成公比的等比数列•前n项和及其性质11111111,(1)1,111111nnnnnnnaqqSaqaaqaaqaaqAqAqqqqqq2322322SSS,SSnnmnmnnnnnmmmmmSSqSSSSSSSnqS偶奇、、成等比数列性质若项数为,则,成等比数列na与的关系:(检验是否满足)nS111;2nnnSnaSSn1a1nnnaSS求通项公式常见方法•(1)观察法;待定系数法(已知是等差数列或等比数列);•(2)累加消元。累乘消元。•(3)•(4)化为构造等比。•化为,分是否为1讨论。1(),nnaafn1(),nnafna1111,()nnnnnaakakaa倒数构造等差:11111,(1)nnnnnnaaaaaa两边同除构造等差:1,nnakab1()()nnaxkax11,1nnnnaqapnraxnyqaxny(构造等比数列:)1nnnaqap111nnnnaaqpppqp