计算(裂项、换元、通项归纳)

1、计算（裂项、换元与通项归纳）:第一部分裂项:11计算11+21+3+4+261220=(1+2+3+20)【2】12(1-12(1-丄21+20丄420111+61220+5120)2021-丄)2120=210-21152_1172-1192-11112-11132-1111111=+一+2446688101012121411,1111,11,11,11=(-+-+24466881010121214=(1-丄)X12142X1+(12+15+(12+15+（13171414丄)5丄)411+(1+丄)5611+(-+1+33+(1+6+-)6【3】计算613=X-14214365,7,9,1

2、1,13+57612203042【4】计算：（此题用到公式一k1）nnknnk231011212123123912310一42345101133661010154555/11、/11、/11、/11、/11、=1-(-)()-(1-()-(-)1336610101545551111111111-1+-+-+336610101545555555【5】计算:9101011101-910113303从这个题目我们可以归纳出一般性的结论。1X2+2X3+3X4+4X5+n(n+1)+(1X3X4X5-1X2X3X4)+33+(1X3X4X5-1X2X3X4)+33=1X1X2X3+(1X2X3X4-1

3、X1X2X3)TOCo1-5hz33311+n(n+1)(n+2)(n-1)n(n+1)331=-n(n+1)(n+2)3即:1223Lnn1!nn1n23另外，例6还有另外一种解法：根据nn12nn所以1223344556677889910121222L9291222L9212L911330-91019-91062.第二部分换元【6】计算:/621739458、)X/739458378、/621739458378、/（一+-+-)+)（一+-+-+)X126358947358947207126358947207(739+458)358947解：设a=型+生8一739458378b=+-358

4、947207358947原式=(621+a)b(+b)a126126【7】计算:【7】计算:+丄2008解：设a621126621126.621b+ab-126x型二920711(1+丄+丄+一31+aab21X(丄+211=一+一+2312007)2007+1+32008120072008原式=(1+a)xb(1+b)xa=b+abaab12008-第三部分通项归纳100100TOCo1-5hz【8】计算：1+丄+-+112123解：先推导出通项公式。1aan=原式=+Z+_+1223341232先推导出通项公式a-nnnnn2、3、4、n1(n1)(n1)2222一X早X斗X.X-9L21

5、31419912233449999-XXXX13243598100299X110099502先推导出通项公式a-nnnnn2、3、4、n1(n1)(n1)2222一X早X斗X.X-9L2131419912233449999-XXXX13243598100299X1100995099+=2X/1111、(+-+)1223341+100101=2X(1-1+1-1+111+)22334100101=2X(1-丄）101=2X10020010110122222/X4XX9922X222131419912100101【10】计算15+Z9,11十13十1517十196122030425672901(1

6、+1)+（丄十1)(1十1)十(1十1)(-十丄）2334455667十（丄十1)(1十1)十(1十-)788991011,1,111,1,1111111,1-11+十-十十-一十十十十23344556677889910+丄1011】计算:+12+101113142+4+1234523456154+164+123452345615,16+1234523454.4+1234523456+2343454564.4345672743456727+6345434567丄12+121011121314101441011121314+-10141011124+和）+672111213)+(12345+23

7、456+34567101112131411111111.=X一+22334344545561112121311.11.112341+23452345|111=丄X+2231213-11,1P1_J5_616【12】计算:22222222221_12.+1231234333I31121333323123343+22+1213312226_32622归纳通项公式an=学2122n3n222221_+12333I3311212211二2X(1+1)31221二2X(1-)327n(n1)(2n1)622n(n1)421-X(-+3n23-1234+333331123422-32226-3261111

8、(1+丄)+(1+1)2334，寸o&HU)is寸寸eeCXICXIL疤LUU1eCXICXILss-1LOLOOL19寸寸22CXI,9Lg寸99C0C09X-I+寸x-;-+(T+T+T+-I)9寸9ggco9寸寸009寸T+T+T+u+-l+-l+T+-r+-THCXILoco9L6+IT+OCXICXILg寸+8C0CXI寸9LJ+J+0000+00+L|irL1X-1+oX+LD1L|OL11CX10-1oXXLOXXX-SH00刖X99LLNCXILLCOOLCXIllco宀-lx-j-I+T-I+T-I二xggLLHgco寸CXIllol9gg寸寸co+1+FxggllHLLLD

9、LL二6LL0Lgcog寸寸CXI寸co十+T+T+T+T金二“LL0L60L689寸co寸COCXI2+二+-+1X99LL(Z)(CXIU)U(CXIU)(Lu)(CXIU)(Lu)UL+lH(Lu)uHue慝fflss(-)LL0L60L689寸co寸C0CXI+X99LLt6lzllg如卄o匸s卜CXICOHXIHCXIg9CXICXI-1001011022334339817、(10.120.23)(0.120.230.34)(10.120.230.34)(0.120.23)解：令1解：令1,，贝U(1(10.120.23)(0.120.230.34)(10.120.230.34)(0

10、.120.23)1118、一+2242461118、一+22424618101681012解：先推导出通项公式。_1_12n2nn1nn12原式_丄+丄+丄+丄+丄+_1-122334455667“1,11+1111,11,11_1丄+丄+_-+丄一+223344556677解：先推导出通项公式。“111119、一+1+335357357211aan_3571_1(2n1)2n13n2nn2132435469111012111/11(-+)+(-1X(1丄)+1X(11)21122121、/10,1、/5_-X+X211212175264132435469111012111/11(-+)+(-1X(1丄)+1X(11)21122121、/10,1、/5_-X+X2112121752