数列、极限、数学归纳法

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0pef[R_³~ÕlN0€²~‰Bl1.Œcácÿ`$ŒcácI{î]peR0I{ÔkpeR„v‚iõ_0y˜lQ_0MRny˜ŒTlQ_ÿa$ý€YÐ(uÙ›NåwƋ㉳QN›Nž[E–˜ÿb$Œcác�gP–„vÛVRÐ�—{ÕlR.2.tã‰ÿ`$peR„vgsQ‚iõ_ÿa$ý€9hnc�¨clQ_—{úQpeR„vMRàQy˜ÿb$OBllQÔk„vÝ~ù[P\ŽN1„vàewzI{ÔkpeRMRny˜ŒT„v�gP–.3.†Nã‰ÿ`$†N㉨clQ_/fÙ~úQpeR„vNÍy¹eÕlÿa$†Nã‰peR�gP–„vaINÿb$†Nã‰pef[R_³~Õl„vŸStÿv^ý€(upef[R_³~ÕlÁ‹fN›N€{US˜.ŒN0åwÆ‹Ó~„g(N)peR„vN,‚‚iõ_peRïSåNw\OåNê�6qpeÆ–(bƒ[„vP[Æ–):NvQš[INßW„výQpeÿàVdkïS(uýQpe„v‰¹p¤‹Æ‹peRÿ(uxvzýQpe„v¹eÕlegxvzpeR0peRhˆ:yÕlgÿRhˆÕl0þVÏPÕl0ã‰�gÕl
0�¨cÕlI{0RhˆÕlÿ1\/fŠbpeR™Qba1,a2,a3&&an&&b€{™Qb[ÿan]ÿÿvQ-Nanhˆ:ypeR,{ny˜„vpePÿn1\/fƒ[„vy˜peÿsSan/fn„výQpe0ã‰gÕlÿ‚YœgpeR„v,{ny˜ý€(uy˜pen„výQpe_hˆ:y:Nan=f(n)ÿÙÍyhˆ:yÕl1\/fã‰�gÕlÿÙ*Nã‰g_ëSZPpeR„vy˜lQ_0þVÏPÕlÿ(WôvÒ‰PWhû|-NÿpeRïSåN(uN¤Rce„vd[Ëz„v¹peghˆ:yÿvQ-NÏkN*N¹p(n,an)„v*jPWhnhˆ:yy˜peÿµ~PWhanhˆ:yå‹y˜„vP0(uþVÏPÕlïSåNôv‰„vŠbpeRanNn„výQpesQû|hˆ:yúQeg0¨cÕlÿpeRïSåN(u$N*NagöNÓ~Tweg„v¹eÕleghˆ:yÿ`$Ù~úQpeR„vNy˜bàQy˜0a$Ù~úQpeR-N(uMRb—„vy˜eghˆ:yTb—„vy˜„vhˆ¾�lQ_ÿÙ/fpeR„vÈSNÍyã‰gÕlhˆ:yÿðy:N¨cÕl0‹O‚YÿpeR2ÿ4ÿ5ÿEMBEDEquation.3
ÿEMBEDEquation.3
&¨cÕlhˆ:y:NEMBEDEquation.3
ÿvQ-Nan+1=an+EMBEDEquation.3
ÈSðy:Nå‹peR„v�¨clQ_01upeRy˜pe„vgP–ŒTàeP–egRpeRSìbwzpeRŒTàewzpeR01upeRy˜Ny˜KNô•„v'Y\sQû|egRpeRSìbžXpeR0ÏQpeR0Fd¨RpeRåNÊS8^peR01upeRTy˜Ý~ù[P„vÖSPƒôVegRpeRSìbgLupeRŒTàeLupeR0y˜lQ_/fxvzpeR„vN*NsQ.•ÿR_³~y˜lQ_/fBlpeRy˜lQ_„vgúW,g¹eÕlÿÙ~úQpeR„vMRny˜ÿBlÙ*NpeR„vy˜lQ_v^N/f/UN„vÿ_Nv^^—@bg„vpeRýý€™QúQy˜lQ_0peR[ÿan]ÿ„vMRny˜ŒT/fÿa1+a2+a3+&+anÿ°‹\OSnÿ‰cknx¤‹Æ‹peRMRny˜ŒT„v&{÷SÿSn/fNÒ‰xn„výQpe0peR„vMRny˜ŒTNy˜KNô•„vsQû|/fan=EMBEDEquation.3
,gUSCQ`N˜˜;N‰g$NÍy{|‹Wÿ`$ò]åwpeR„vy˜lQ_b¨csQû|™QúQpeRbpeR„vÐgNy˜0ÐgàQy˜0a$1u˜˜¾‹™QúQpeR„vy˜lQ_0(ŒN)I{î]peRŒTI{ÔkpeR1.I{î]peRš[IN0hˆ:yÕlÊS'`(�(1)I{î]peRš[IN-Nÿ‰ÆQnx0Wtã‰ÿ3zeP0W”^(ulQî]dÿÆQnx„vtã‰sSèlaš[IN-NÎN,{ŒNy˜w�ÊSTN*N8^pe„v+TINÿ3zeP0W”^(usSan+1-an=d/fÁ‹fpeR/fI{î]peR„vtº‹�OncKNNÿ€d„v&{÷SÈS³Qš[I{î]peR„vUSŒ'`0(2)‚YœgN*NpeR[ÿan]ÿ/fI{î]peRÿlQî]:NdÿRÙ*NpeRïShˆ:y:Nÿ`$RhˆÕlÿa1,a1+d,a1+2d&a1+(n-1)d&€{™Qb[ÿa1+(n-1)d]ÿÿyrŠk0WÿêSgNy˜öeïS™Qbÿa-d,a,a+dÿêSgÛVy˜öeïS™Qbÿa-3d,a-d,a+d,a+3d.hˆ:yĉ‹_ÿGYpey˜lQî]:N2dÿvPpey˜lQî]:N2dÿƒ[ìN/f㉳QI{î]˜„v¡‹—{å]wQ0a$ã‰�gÕlÿan-an-1=d(ne2,nN)ÿyrŠk0WÿêSgNy˜öeïS™QbA-x=y-AÿsS2A=x+yÿvQ-NAëSZPx
0y„vI{î]-Ny˜ÿƒ[ìN/f㉳QI{î]˜„vÁ‹få]wQ0b$þVÏPÕlÿan=a1+(n-1)dïS9e™Qban=dn+a1-dÿÙhˆfS_d`0öean/fsQŽNn„vN!kýQpeÿàVdk(WôvÒ‰PWhû|-NI{î]peR„vþVÏP/fÿåNd:Nœe‡sÿ(Wyt�N*bÝ�:Na1-dÿv^Nn:Nê6qpe„vNagôv¿~NN›NRce„v¹p0(3)I{î]peR„vy˜lQ_ò]åwa1ŒTlQî]dÿRgan=a1+(n-1)dò]åwamŒTlQî]dÿRgan=am+(n-m)d(m,nN)(4)I{î]peR„vMRny˜ŒTlQ_ò]åwa1ŒTanÿRgSn=EMBEDEquation.3
(nN)ò]åwa1ŒTdÿRgSn=na1+EMBEDEquation.3
d(nN)(5)I{î]peR„v'`(�`$(WI{î]peR„vMRny˜-NÿN$NïzI{Ý»y„v$Ny˜KNŒTGWøvI{ÿsSÿa1+an=a2+an-1=a3+an-2=&&=ar+an-r+1=&&a$(WI{î]peR-Nÿå‚Ðg$Ny˜„vy˜peKNŒT/fš[PÿRøv”^„v$Ny˜„vpePKNŒT_N/fš[P0sSÿ(WI{î]peR[ÿan]ÿ-Nÿ‚Yœgm+n=p+q(m,n,p,qN)ÿ£�HNÿam+an=ap+aqb$(uþVÏPÕlhˆ:yI{î]peRöeÿvQT¹pGW(WåNlQî]d:Nœe‡s„vNagôv¿~NÿsSd=EMBEDEquation.3
(m,nN,m`n)c$I{î]peRI{Ý»y„vÖSúQå‚r^y˜ÿÍN6q/fI{î]peR0d$lQî]:Nd„vI{î]peRÿcky˜RÄ~ÿÏkky˜KNŒTÄ~b„vpeRÍN/fI{î]peRÿvQlQî]:Nk2d.sSÿa1+a2+a3+&+ak,ak+1+ak+2+&+a2k&ank+1+ank+2+&+ank+k&ÍN/fI{î]peR0e$$N*NI{î]peR„v,{ny˜KNÔkI{ŽNMR2n-1y˜KNŒT„vÔk0f$peR[ÿan]ÿbI{î]peR„vEQ‰agöN/fan=dn+c(d,c:N8^peÿnN)g$peR[ÿan]ÿbI{î]peR„vEQ‰agöN/fSn=an2+bn(a,b:N8^peÿnN)2.I{ÔkpeRš[IN0hˆ:yÕlÊS'`(�(1)(WI{ÔkpeR„vš[IN-N‰ÆQnx0Wtã‰ÿup;m0W”^(ulQÔkqÿÆQnx0Wtã‰sSèlaš[IN-NÎN,{ŒNy˜w�ÊSTN*N8^pe„v+TINÿèlalQÔkqNý€:Nö–0up;m0W”^(uhˆ°s(WÿS_EMBEDEquation.3
=q(nN,q:N8^pe)öeÿdkpeR/fI{ÔkpeRÿhˆ°s(WS_q-ÿ0öeI{ÔkpeRTy˜&{÷SGWøvTÿS_qÿ0öeTy˜&{÷Sck�øvô•ÿhˆ°s(WS_\ÿq\ÿ-ÿ1öepeRÏkNy˜ÖSÝ~ù[PT/fžX„vÿS_\ÿq\ÿÿ1öepeRÏkNy˜ÖSÝ~ù[PT/fÏQ„v0(2)‚YœgN*NpeR[ÿan]ÿ/fI{ÔkpeRÿlQÔk:Nqÿ£HNå‹peRïShˆ:y:Nÿ`$RhˆÕlÿa1,a1q,a1q2&a1qn-1&ïS€{™Qb[ÿa1qn-1]ÿyrŠk0WÿêSgNy˜öeïS™QbÿEMBEDEquation.3
,a,aqêSgÛVy˜öeïS™QbÿEMBEDEquation.3

0EMBEDEquation.3

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0aq0hˆ:yĉ‹_ÿGYpey˜lQÔk:Nq2ÿvPpey˜lQÔk:Nq2ÿƒ[ìN/f㉳QI{ÔkpeR˜„v¡‹—{å]wQ0a$ã‰�gÕlÿEMBEDEquation.3
=q(nN,q`0)yrŠk0WÿêSgNy˜öeïS™QbEMBEDEquation.3
=EMBEDEquation.3
bG2=xybG=±EMBEDEquation.3
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