Bồi dưỡng toán 11 nâng cao phần dãy số

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.Chứng minh phương trình : x = x + 1 có nghiệm Giải Đặt f(x) = x – x – 1 thì f(x) liên tục trên RTa có : thì  ít nhất x  0 ; 1  : f( x ) = 0Vậy phương trình : x – x – 1 = 0 có nghiệm 4.Chứng minh phương trình : x –3x = 1 có ít nhất một nghiệm thuộc  1 ; 2( f(1).f(2)< 0 )5.Chứng minh phương trình : m( x – 1)³.( x + 2 ) + 2x + 3 = 0 luôn có nghiệm ( f(1).f( – 2) < 0 )6.Chứng minh phương trình : a( x – b )( x – c ) + b.( x – a )( x – c ) + c.( x – a)( x – b ) = 0

                       dUU nn + = +1  !" #$%&' () ( ) dnUU n 1 1 − + =   2 11 +− + = nn n UU U   ( ) ( ) [ ] 2 12 2 1 1 21 ndnU nUU UUUS n nn −+ = + =+++=  &* * qUU nn = +1  * () 1 1 − = n n qUU   11 2 +− = nnn UUU  () ( ) 1 1 1 21 − − =+++= q qU UUUS n nn  ++, - +, $./0!123 45 37;2 81 = = UU  +, -&67890: ;8!12<19 nnS n 65 2 +=  ! 45 n S ,12 =>?97@/ n U 812AB$C 1 − − = nnn SSU #$DE$.F G ( ) ( ) [ ] 101110110 1 = + − − + = − − nnUU nn #1 H-12026,12  1#$D +, -I631JK<735&$,96;!L3$M9% ;!$,69  +, $NO78$&90: ;8!<90: ;,IPO8 9I,Q35&R ?H 1# 1 2 − S? $& #TU H 1# 5 2 S? $& #$&U +, &NS!/98!3 4578R0: ;35RDP78R  V35$DRN78W/935&$T =>X WY12WZS!, ( ) n a [ 0<\6 W  1 2 3 4 1 2 3 40 104 n n n n a a a a a a a a − − − + + + =   + + + =   ] ^_/ 1 2 1 3 2 4 3 n n n n a a a a a a a a − − − + = + = + = + >60< ( ) 1 4 144 n a a+ = 1 36 n a a  + =  [ 0< 1 216 12 2 n n a a S n n + = =  = >6- 1 1 4 6 40 4 38 104 a d a d + =   + =  1 7 2 a d =  ⇔  =   H- ( ) n a ÷ < ( ) 1 7 a = ,  2 d =      +, INS!/ /48 α 0` 2 1 sin ;sin ;1 sin 3 α α α + + -, ?6 α #Da 1 sin 2 α − =  +, R! 4512<97@/ 2 5 n a n = − -,NS!? &D  ?1# 1 2 n n a a − − = , 20 320 s =  +, b+1JK,R ; 8!< 1#&,O8c35 $UIDbNS!/0< X" RWZS!, ; 2; 4; 6 a a a a + + + N\6 W  ( ) ( ) ( ) 2 4 6 19305 a a a a+ + + = ( ) ( ) 2 2 2 6 6 8 19305 a a a a + + + =  0^ 2 6 u a a = +  [ 0< 135; 134 u u = = −  • H  2 6 135 9 15 a a a a + =  = ∨ = −  • H  2 6 134 a a+ = −  G! +, TN46!< 1 n a m = , 1 m a n =   ( ) m n ≠ =2S!78!90: ; 812 =>N< 1 1 a d mn = = 1 1 2 2 mn mn a a mn S mn + +  = =  +, -6 1 2 ; ; n u u u 460< 0 1, i u i n > ∀ = , 1 $ ! 45 1 2 2 3 1 1 1 1 1 1 . . n n n n u u u u u u u u − − + + + =  & ! 45 1 2 2 3 1 1 1 1 1 1 n n n n u u u u u u u u − − + + + = + + + +  I ! 45 1 2 1 1 2 1 1 1 2 1 1 1 1 2 1 1 1 n n n n n n u u u u u u u u u u u u u − −   + + + + = + + +   +    =>$X" ?# 1 2 2 3 1 1 1 1 . . n n u u u u u u − + + +  _ 0<1?# 1 2 2 3 1 . . n n d d d u u u u u u − + + + # 3 2 1 2 1 1 2 2 3 1 . . n n n n u u u u u u u u u u u u − − − − − + + +  1 1 2 2 3 1 1 1 1 1 1 1 1 1 1 1 n n n n n u u u u u u u u u u u u − − = − + − + + − = − = ( ) 1 1 1 1 n n n d n dS S u u u u − −  =  =  H- 1 2 2 3 1 1 1 1 1 1 . . n n n n u u u u u u u u − − + + + =  =>& 3 2 1 2 1 2 1 3 2 1 n n n n u u u u u u S u u u u u u − − − − − = + + + − − − >6 2 1 3 2 1 n n u u u u u u d − − = − = = − =  ?4?# ( ) ( ) ( ) 1 1 1 1 1 1 1 n n n n n d n u u u u n d u u d u u d u u − − − − = = = + + +      =>Id:! JK0JK ( ) 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 ( ) 2 n n n n n n u u u u u u u u u u u u u − −   + + + + + = + + +     A(C X"  S = ( ) 1 1 2 1 1 2 1 1 1 1 1 ( ) n n n n n u u u u u u u u u u − − + + + + + HS46!" < 1 2 1 1 2 1 n n n n u u u u u u u u − − + = + = = + = + >60<?# 1 2 1 1 2 1 1 2 1 1 2 1 ( ) n n n n n n n n u u u u u u u u u u u u u u u u − − − − + + + + + + + +  1 2 1 1 2 1 1 1 1 1 1 1 1 1 n n n n S u u u u u u u u − − ⇔ = + + + + + + + + # 1 2 1 1 1 2 n u u u   + + +     H-A(C0c +, -! 4546!" < ( ) 3 2 3 n n n S S S = −  N--He# ( ) 2 3 n n S S − ( ) ( ) ( ) ( ) 1 1 2 2 1 2 2 1 3 2 2 u n d n u n d n   + − + − = −       # ( ) 1 3 2 3 1 3 2 n u n d n S + −     =  +, -662!2O m n S S =   m n ≠ ! 45 0 m n S + =  =>Nf W  ( ) ( ) 1 1 2 1 2 1 2 2 m n u m d m u n d n S s + − + −         =  = ( ) ( ) 2 2 1 1 2 2 u m m m d u n n n d  + − = + − A(C ( ) ( ) 1 2 1 0 m n u m n d  − + + − =     >6 m n ≠ ;fA(C4 ( ) 1 2 1 0 u m n d + + − = A((C ] ^_/ ( ) ( ) 1 2 1 2 m n u m n d m n S + + + − +     = H- 0 m n S + =  +, -N7890: ;812, 3 1 n n S = − ! 4512,12 * =>NfO7812 ( ) ( ) 1 1 3 1 3 1 n n n n n a S S − − = − = − − − 1 1 3 3 2.3 n n n n a − −  = − =  .FJK 1 2 1 3.3 3.3 n n n n a a − − − = #I ,g { } n a ,*<3 @#I +, -S!I<735$&R,I90: ; 8!*,0hi ,/ 9I%$I%$b8! =>X WY 1 3 2 13 3 15 ; ; u a u a u a = = =  [ 0< 1 1 2 1 3 1 2 ; 12 ; 14 u a d u a d u a d = + = + = + H 1, 8 N\6 W  1 2 3 124 u u u + + = ( ) 1 3 28 124 1 a d + =  ] ^_/N\6O8* ( ) ( )( ) ( ) 2 1 1 1 1 12 2 14 2 29 0 2 a d a d a d a d+ = + + ⇔ + =  NfA$C,A&C4 1 116; 8 a d = = − H- 1 2 3 100; 20; 4 u u u = = =  +, -63-,*Lc4fJK6&%$aQa&Qa-0J j!=2S!/0< =>X" /WZS!, 2 3 ; ; ; a aq aq aq  [ 0<\6 W &a@&a@ & kQa@ I k&Q-,>60< ( ) ( ) ( ) 2 2 1 2 7 aq a aq − = − + −      ( ) ( ) ( ) 2 3 2 7 1 27 aq aq aq− = − + − ?4 ( ) ( ) 2 2 1 7 1 14 a q aq q  − =   − =   7 2 a q =  ⇔  =   H-3WZS!, +, -N7/98!*V 935bT<73SJK8/ 98<35RRl=2S!, =>N\6 W  1 1 1 1 56 1 n a a a q a q q + + + = = − , 2 2 2 2 2 1 1 1 1 2 448 1 n a a a q a q q + + + + = = −  [ 0<< ( ) 1 2 2 1 56(1 ) 448 1 a q a q = −    = −   ( ) 1 8 1 a q  = + >60< ( ) ( ) 3 8 1 56 1 4 q q q + = −  = , $ #$R +, -+<735Tb-,!*mL3!0KM, 3$U0KM30Jn!NS!I0< 3NS!3-,*3 4578c35&TP7/3SJK8 c35ITR ?ba$baRb +$laTa&6^&aTa$l +, -6* 1 2 ; ; u u ! 45 2 2 3 2 n n n n n n n S S S S S S S − = − −  N--HN# ( ) ( ) ( ) 1 2 2 1 1 1 1 1 1 1 1 1 1 1 n n n n n n u q q q q q u q u q q q − − − = − − + − − − − − # ( ) 1 1 1 n n n n q q q q − = − A(C He# ( ) ( ) ( ) ( ) 2 1 1 2 3 2 3 2 1 1 1 1 1 1 1 1 1 n n n n n n n n u q u q q q q q q u q u q q q − − − − − = − − − − − − # ( ) 2 2 1 n n n n n n q q q q q q − = − A((C H-fA(C,A((C4e] +, -! 45 ; ; 2 2 2 A B C tg tg tg N\6o-,S6pa6+a6 q\6o-, =>N\6 W  sin sin 2 2 2 2 2 2 2 cos cos cos 2 2 2 A C B A C B tg tg tg A C C + + = ⇔ = 2 cos 2sin cos cos 2 2 2 2 B B A C ⇔ =  1 cos 1 cos cos cos B B A C ⇔ + = − + + 2cos cos cos B A C ⇔ = +  +, -! 45 ; ; 2 2 2 A B C cotg cotg cotg N\6o-,S39a 3aq\6o-,     =>N\6 W  sin cos 2 2 cot cot 2cot 2 2 2 2 sin sin sin 2 2 2 A C B A C B g g g A C B + + = ⇔ = 2sin 2 cos 2 A C A C + = + sin cos sin cos cos 2 2 2 2 2 A C A C A C A C A C + + + − +   ⇔ = −     ( ) ( ) ( ) 1 1 sin sin sin sin sin sin 2 2 A C A C B A C ⇔ + = + ⇔ = + 2 ; ; a c b a b c ⇔ + = ⇔ ÷  +, -! 45 ; ; 2 2 2 A B C cotg cotg cotg N\6o-,S39 2 2 2 ; ; a b c q\6o-, =>N\6 W  ( ) sin 2cos cot cot 2 cot sin .sin sin A C B gA gC gB A B B + + = ⇔ = 2 2 2sin 2sin 2sin sin cos B B C B ⇔ = 2 2 2 2 2 cos b ac B a c b ⇔ = = + − 2 2 2 2 2 2 2 ; ; b a c a b c ⇔ = + ⇔ ÷                                                                   ∀ ∀∀ ∀  !"#$%& "' ' ()& * r  − +  /FO  9eF%4f%F*%   /O8  9%;_s     C] t 12<97@/ r  # Q  I + S   ru ! +∞→ #I 3C612Ar  C r  # b  &  & & + + ] t   ru ! +∞→ #D =>  D r − # b  &  & & + + v &  & #  & vB  & " D # $  & +        C612Ar  C r  # R  &  & & & + + + ] t   ru ! +∞→ #$ 6@!, @ v$] t   @u ! +∞→ #D C] t =−+ ∞→ C$Au !  Da 3C] t D $  ? & u ! I  = + ∞→    !"#$%&%'(') *NS!/  9  C $$   IC&A IC&A u ! ++ ∞→ +− +− a3C C$bAu ! &&  +−+ ∞→  +NS!/  9  C & I& u ! & &  −+ ++ +∞→ a3C $u ! &  −++ +∞→  C ( ) $ $  I& $ &$ $ u !  + +++ +∞→ a1C R &&&   C$A&I$& u ! ++++ +∞→  =>C WY,!w6 & 3CL* ;n Cr  #$ $  $ + >60<u !r  #$ ! $  $ + #$ , C612Ar  Cx/0J r  # C&CA$A $  RI& $ I&$ $ ++ +++      NS!   ru ! +∞→  3C612Ar  Cx/0J r  # I& & $&  & b & I & $ − ++++  NS!   ru ! +∞→  - 612<Or E$ k&r  Er $ #λH λ,!5 =2O  9 &    r u ! +∞→  ,-./ 0 +, $C3FVN< 07 7 21 7 21 3 7 3 3 >+ + = + =− + =− ndo nnn n Un  >6- Ir  − v ⇔ Q  &$ + v⇔B Q  &$ − M0* D #       − Q  &$ E$[ 0<fA$C 4∀  D S Ir  − v   ru ! +∞→ #I C R& R $ R& & $r && &  ++ =− ++ + =−  >6-  C$A R  R& R $r &&  < + < ++ ⇔<− B $  & −   D # $$  & +       −  +, &>6 @ v$ $ @ $ > S∃ $ @ $ +=  N< $ $ C$A $ @D@   + ≤ + ==−  Ac+N+\ A$EC  H 4JxF+0 C$  $ A  $   $ $   $ $ −>>+⇔< +  D # $$  $  $ +             −  +, I C $ $ $$Dr  ++ =−+=−+=−   &   R $  & $ 6 & $ Dr ><<−  D # $ R $ & +            3C &II   & $  & ?  $  & Dr < + < + =− 6 &  & vεB  &  D # $  & +  +, RCu $ # $$   IC&A IC&A u ! ++ ∞→ +− +−  # $ I & I $ I & I $ u ! $   +       − +       − + +∞→  N] 0Jn D I & u ! I & u ! $    =       =       − + +∞→+∞→  /1y0/FO_   9<u $ # $ I $ # I $  3CL* ;n%/1y = +∞→  R u !  = +∞→ &   b u ! = +∞→ &   $ u ! DN0Jnu & #D +, b1CN< &$ & EI& & E'EAE$C & #A$E$C$ & EA&E$C& & E'EAE$C &  #A$ I E& I EI I E'E I CEA$ & E& & EI I E'E & C# T C$&CA$A R C$A && ++ + +  R &&&   C$A&I$& u ! ++++ +∞→ # R &&   R C$A u ! + +∞→ E R   T C$&CA$A u ! + + +∞→ # R $  +, TC &_ $ $_ & _ $ C&_CA$_A_ $ + + + −= ++ r  # $  $ &  $ & $ + − + +  ?4 !r  #$)& 3Cr  # I& & $&  & b & I & $ − ++++ A$C?4 =  r & $ $RI& & $&  & b & I & $ + − ++++ A&C N4ff8A$C,A&C0Jn =  r & $ $I& & $& & $  & $ & $ & & $ + − −         ++++  # $ $ & & $& $ & $ $ & $ & $ & & $ + − − − − −       + ?4r  #$E& $ $ & $& & $ $ + − − −               − AIC >6 D & $& u ! & $ u !   $  = − =       +∞→ − +∞→ H-AIC_ /1y/O  9 <   ru ! +∞→ #D     +, Q Nfr E$ k&r  Er $ #λ4Ar E$ r  C#Ar  r $ CEλ ^H  #r  r $  Nf4;<H E$ #H  Eλ0 0<<AH  C-,  λ N<r  #r $ EAr & r $ CEAr I r & CE'EAr  r $ CA$C#r $ EH & E'EH    H & EH I E'EH  # [ ] & C$AC&AH& & − − + #Ar & r $ CA$CE C&IA &  & +− A&C NfA$C,A&C4 r  #r $ EAr & r $ CA$CE C&IA &  & +−   & &  &I & $& & $ &  &   $ CrrA  r  r +− + − −+= AIC_ /1y/O   9< &    r u ! +∞→ # &                           p,  98=,!zAxC_ x  x D J !"  εBDh9 δ66∀x!, D xx − vδS CpCxAz − vε cx≠x D  . ,12 =>?97@/ n U 812AB$C 1 − − = nnn SSU #$DE$.F G ( ) ( ) [ ] 1 0111 0110 1 = + − − + = − − nnUU nn #1 H-12026,12  1#$D +,. 1 1 21 ndnU nUU UUUS n nn −+ = + =+++=  &* * qUU nn = +1  * () 1 1 − = n n qUU   11 2 +− = nnn UUU  () ( ) 1 1 1 21 − − =+++= q qU UUUS n nn  ++, - +, $./0!123. dUU nn + = +1  !" #$%&' () ( ) dnUU n 1 1 − + =   2 11 +− + = nn n UU U   ( ) ( ) [ ] 2 12 2 1 1 21 ndnU nUU UUUS n nn −+ = + =+++=  &* * qUU nn = +1  * () 1 1 − = n n qUU

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