Bồi dưỡng học sinh giỏi khảo sát hàm số

510.76 B452D NGUYEN TAI CHUNG (B6i dtedfig hoc sinh gioi Todn, chuyen Todn) T A I B A N C O StTA CHUA V A B O SUNG Phan loai toan day so Phuong phap giai toan day so ^ Cac de thi hoc sinh gioi Quoc gia, khu vac -if Cac de thi Olympic Sinh vien, Olympic 30/04 Tim day so {Xnlnri c h o x i = a va xn.i = a x ^ b x ^ c X n + d,VneN* f a > 0,c= y DVL.013494 Tim so hang t o n g q u a t cua day so da cho d = ^ ^ - I f ^ ) ia z/a^ (1) NGUYEN TAI CHUNG B6i dvtdng hoc sinh gioi, chuyen toan C H U Y E N D E D A Y s6 V A I ? N G D U N G - Phan loai toan day s6 PhiWng phap giai toan day s6 Cac de t h i HSG Quoc gia, K h u vuTc Cac d l t h i Olympic Sinh vien, Olympic 30/04 J*' N H A X U A T H A \I I I Q C Q U O C G I A H A NOI -1 ^I3 (ivj*-! 1" Chu'dng L&i noi dau Day so la mot chuycn de quan trpng timoc clutrtng trinh chiiycn toan c&c trUdng THPT chuyen Cac bai toan lien quan den day so thiTcJng la nhufng bai t$p kho, thirdng g?ip cac ki thi hpc sinh gi6i m5n To&n cap qu6c gia, khu v\fc, quoc te, Olympic 30/04 va Olympic Sinh vien Toan day so rat phong phu, da d^ng va cung rat phutc h^p nen kho phan lo^i va hg thong hoa cac chuycn ricng bigt Tuy v^y, chuug toi c6 gang toi da sSp xep hpp li de giiip ban dpc tiep can tijfng bifdc, ttoig miic dp kien thiic va luyf n t§.p giai toan Phan 16n cac bai toan cuon sach du^c tuyen chpn t\t cac ki thi; Thi hpc sinh gioi quoc gia, thi chpn dpi tuyen quoc gia dil thi toan quoc te, Olympic 30/04, Olympic toan Sinh vien toan qu6c, thi hpc sinh gioi ciia cac ti'nh thanh, ho$c tren T^p chi toan hpc va tu6i tre Mpt so bai toan khac la tac gia t\f bieii soan Xiii cam dii tac gia cac bai toan nia chiing toi da tri'ch chpn Nhiing Idi giai dua dua tren tieu chi tir nhien, de hieu 'l\iy nhien, IcJi giai d day chua han la Idi giai hay nhat va ngSn gpn nhat, hi vong ban dpc so c6 dxtac nhftng Idi giai hay hdn Vice phan chia cac chiroug, bai, myc dfin c6 tinh dpc lap tiWng doi va do khong nhat thiet luc nao cung phai dpc theo trinh t\i Cung can noi them rSng, that kho nia i)han chia cac van d^ theo mot bien gidi rach roi nhit tieu dg ciia tCrng bai Dau mpt vai van de ciia bai da xult hien bong dang van dg ciia bai Tuy vay, chiing toi van c6 gang xoay quanh chii de ciia bai iy de ban dpc minh rut TiliuTng gi thudng gap va each giai quyet van de Tot nhat, dpc gia t\t minh giai cac bai tap co sach Tuy nhien, dc thay va lam cliii nhiiiig ki xao tinh vi khac, cac bai tap deu dUcJc giai san (tham chi la nhilu each giai) v6i nhiJng mute dp chi tiet khac Npi dung sach da c6 gang tuan theo y chii d^o xuyen suot: BiSt diftfc IcJi giai ciia bai toan chi la yeu cku dau ticn - ma hon the - lam the nao de giai ditdc no, each ta xii li no, nhulig suy lu^n nao to "c6 li", cac ket luan, nhan xet va litu y tiT bai toan difa Hi vpng quygn sach la tai lieu tham khao c6 ich cho hpc sinh cac 16p chuyen toan trung hpc ph6 thong, giao vien toan, sinh vien toan ciia cac trirdng DHSP, DIIKIITN ciing nhir la tai lieu phyc vu cho cac ki thi hpc sinh gioi toan TIIPT, thi Olympic loan Sinh vien giiJta cac tritdng dai hpc Xac dinh day so 1.1 1.1.1 Xac dinh day so bang phu^dng phap quy nap, phu'cfng phap doi bien X a c d i n h d a y so b a n g phifdng p h a p q u y n a p chiing minh menh de chiia bien A{7i) dung v6i moi so nguyen dUdng n (bang phifdiig phap quy nap), ta tlittc hieu ba bitdc sau: Bxidc {hxidc cd sd, hay bvTdc khdi d l u ) Kiein tra A{n) diing n = Biidc (birdc quy n a p , hay bi:rdc "di t r u y § n " ) Vdi fc e Z, fe > 1, gia sii A{n) diing n ^ fe, ta chiing minh A{n) cung diing n = fc + Bifdc Ket luan A{n) diing xcii moi so nguyen ditdng n B a i t o a n Cho day s6 nhu sau: | = ^ , ^ + ^ , Vn = , , a) Tinh x\, X2, Xi b) Tim so hang tSng qudt {so hang thii n) a) Ta CO x i = \/2 = cos X = V^2 + v / = W2 , va + v/2' = ^ ( + COS J ) = 2cos J , X3 = V'2T1^ - ^ ( + COS J ) = cos ^ Tac gia Th^ic sy Nguyen Thi Chung , b) Vdi e Z, k>l, BSLng phudng pliap quy nap ta se chiing minh gia sii xfc = c o s K h i Un = V2T^k = y ^ l T c ^ ^ ^ = 2COS Theo nguyen l i quy n?ip suy x„ = cos ^^^n TrUdng h^p n = da kigm tra d trgn Gia sil Un = tan [ | + ( n - l ) ^ ] K h i = 1,2, tan x„ = ^ + \/2 + • • • + v ^ Hay tinh xjoro- Bai toan N ^ CO ' n dau cSn x„ = v/2 + x„_i, Vn = 2,3, Un sa\ sir di.ing bai toan d trang Bai toan (Do thi OLYMPIC 30/04/2003) Cho day s6 (u„) djnh bdi U2003 = Tim Un + y/2~ V ^ ) Un ,Vn= = tan L3+("-^)8J = tan tan 1,2, SO J tanx + tany ^ - tanx tan?/ = tan - = tan ( - + ^8 = -2 = = a*+^+-* + t-?"'' yf ,n-l = a ' - y\ = a'^'^'^t^ 3"~'-l I" / 3a J \ 3»-> y „ ^ , - ^ = a ( j , „ - A ) + ( y „ - A ) + c , V n = 1,2, ( V*yx„ = a (^a + - j - - , V n = l,2, Luu y Chd y d trang da gpi y cho ta each doi bien J/n = hin b'^\ f + 3^- b \ Bai toan (De thi HSG quoc gia nam hpc 2000-2001, bang B) Cho day 62 62 62 - 26 -l = cosh 2(f), j/3 = cosh 2^ (A, , 2/„ = cosh 2"~ ^ (A Tif cosh cho cos(^ = y^ Khi = cos 2"~^ • Khi I2/1I > Xet so thi^c fj cho > Neu \yi\>\i ton tai T , = yi li' = 2/1 + ^^y\ Do _ 2" — , j/n = + + (yi + y/yl -1) — , V n = 1,2, B a i toAn 11 Tim day so {x„}+~i x i = Q, x„+i = ox^ + 6, Vn e N ' , at = - Vay neu dat P = yi + i/j/jf - thi 2/1 = Hvtdng d i n Dat x„ = - y „ , yi = / I -6y„+i = at^y^ ^ ^ ^ ^^^^ = _aj,y2 _ y^^^ = 2y2 - Taco Sau Slit dung bai toan 10 Dodo LtTu y Phep dat x„ = - y „ ditdc tim nhit sau: Cong thitc hwng giac C O S Q = 2cos2 Q - gdi y cho ta c6 gang dua day so da cho ve day so {yn)t=i thoa man y„+l=2y2-l,Vn=l,2, (1) D§,t x„ = py„ Khi / py„+i = ap^yl + y„+i = apyl + | Ttf (1) va (2) suy ta can tim p cho f ap = r \a man a6 = - ) V^y ta se dat Theo nguyen h' quy n^p suy /^2"-> 10 ^ = -y„ a x„ = - y „ (do a6 = - ) ,V7i = l , , 14 (2) B a i todn 12 Cho day so (x„) nhu sau: x i < va Vi 4b „ = 2j/„ = 4x„ nen ZxZ2Zz Zn XiX2X3 X„ 4'-"x„+i 4x„+i 4".XlX2X3 X„ So hang t6ng quat ciia day so (w„) la Tim so hang long qudt cua day so {u„) 1\ -, Vn G N* G i a i Dat x„ = ^y„ Ta dUdc day so (y„) thoa man: j / i = 2xi < - va B a i t o a n 13 Ttm day so {x„}+f°i biet XiX2X3 X„ X I = Q , x „ + i = a x ^ + , V n N * , a = Sijf dung bhi toan 10 ci trang 10 ta dildc / vdi /3 = 2/1 + v^y2 _ ^ + \ G i a i D$,t x„ = 6j/„, ,Vn=l,2, = - va = abyl + 6i/„+i = a62y2 + =^ + ^4x2 - Dat y„+i = 2y2 + (do a6 = 2) Xet so th^tc /3 cho jn-l I- Khi Vay neu dat ^ = yi + y ^ y f + T thi Taco Boi vay 2/1 = \ 1 y2 Pj + _ = 2y? + = Gia sii Vn = ^ (^P'^'"' - y„+i = 2yi + = 12 pj\ m +1 = r ) K h i P 13 \ T2 \ -— Tlieo nguyen If quy nap toan IIQC suy B a i t o a n 15 Tim { x , , } ^ ^ , hiet x i = n, 2/n = Xn+\ o x „ - x „ , a > Af „o Bdi v^y H i r i n g d i n Dfit x „ = - p / n - K h i y„+i = Ayl - 32/„, Vn = 1, , Sau 2n-l + a V6^ + ' B a i t o a n 14 Tim {yn}n=i' a ,Vn sii dung bai toan 14 Lifti y Phep doi bien trcnig bai toan 15 d traiig 15 dildc t h n r a n h u sau: T i t h? thiic t r u y h6i ciia day ( x „ ) khien t a hen tucing den rong t h i k ludng giac biet cos Ta C O g^ng yi R, y„+i = Ayi - 3j/„ 3x = C D S ' * X — cos x d y n g cong thi'rc Gia sii x „ = 6y„ + c, k h i bVn+i + c = a {byn + c)^ - (62/,, + c) Giai «>6y„+i + c = a {b'^yi + Sb^cy^ + 36c^2/„ + c^) - (6y„ + c) • Neu 12/11 < t h i ton t ^ i cho cos(j> = j / i K h i / = cos^ (/) - cos = cos , , 2/n = cos 3"0 • K h i I2/11 > X e t so thyc l3 cho Vay t a cho /? = 2/1 + L /? = y i Vay neu dat (3 = -1 «>y„+i = abSi - + 3o6cyf + Z{ac^ - l ) y „ + ^-'(ac^ - 4c) a6^ = 3a6c = 3(ac2 - 1) = - '(ac^-4c) = V2 thi c = 0^ = Do t a dat x „ = —;=2/u- T u y uhien c6 t h ^ t i m cong thiic doi bien nhanh )' Ta -r'''- hon bSiig each dat x „ = byn, sau t i m CO B a i t o a n 16 Ttin {x,,},"^^, biet xi = a, x,i+i = axf, + 3x„, a > y:-2 2/2 = Giasu2/„ = - 2/n+l = G i a i Dat x „ = -7=2/11- K h i 2/1 = ( yn] +3 Vv/a - /fl-^ + : ? r : ^ , -3 ^ Xet so thuc /:* clio 3„-, _ _ Vay theo nguyen h' quy nap toan hoc suy ,Vn = , , 14 va \ ^2/a+i =4yf.+3y„(Vn6N*) / yi = - " Vay nou dat /J = yi + V y f T T • /3 = y i + / y ? + '^^'^ - = ^ J i = y i - x / y f + i thi / yi = v 15 Bai to&n 19 Cho day so (u„) nhusau: | '^^^ = + Su^ - 3, Vn = 1,2,, 7^m cong thiic so hang tong qudt cua day so da cho Ta CO Giai D^t ii„ = u„ + (n = 1,2, ) Khi Hi = v& Do (16 =4 yn+1 = /^a— \ 03" ,Vn = 1,2, Khi T3 \ +3 ^3" l,^ /J Theo nguyeu h' quy n^p suy y„ = ^ / j a - _ x„ = + +1 2\,^ /33"-» ^3"^- Vn = 1,2, V$y - \" + +1 ,d= (1) HUotng d t n Gpi y,, = x„ + ^ , thay vao (1) ta dudc 2/„+i = ay^ + 3y„ o(l Bai toan 18 (De nghi thi OLYMPIC 30/04/1999) Xdc dmh so hang tong qudt cm day so (u„) biet iting: Hifdng d t n f u, = \i = 9uf, + 3(t„,Vn = 1,2, Cach Dat u„ = - X u , xi = ; a;„+i = ^ + 3x„, Vn = 1,2, Den day, ta tien hanli luaug tu nlut bai toaii 16 d trang 15 each Dat 3a„ ^ ^ { ,,,, + 3,,^^ Chon xux^ cho { §| + Bang quy nap, chiing niinh dUdc: v.^, = v l - 3t;„ = ( x f - ' + x f " ) ' - 3{xr-' + x f " ) = xf + 3xf x\f + xj )+xi - Z{x\ x5 ) = x f + 3(xr-' + x f " ' ) + xf - 3(xr-' + xf") = x f + xf fi V$.y theo nguyen ly quy n9,p suy Vn = x?"~* + x^""', Vn = 1,2, V$,y Bai toan 17 Tir/i day S(J {^ulit^ *'ao c7io i i = a i„+, = ax^+6x^+cx„+d,V7i € N* a > , c = — i;„+i - = K - ) ^ + K - 1)2 - = - t ; „ - V$,y Vn+i = ^ n - 3i^n, Vn = 1,2, Den day ta tien hanh tUdng tit nhit bki toan 15 d trang 15 Cach khac Xet phUdng trinh - 3x + = Phuong trinh n&,y e6 hai nghiem xi, X2 va theo dinh ly Viet ta c6 xi + X = vk X1X2 = Ta se chiing minh quy n?ip rang Vr, = x\ , V n = l , , Ta e6 t;i = = xi + X2 = x f + xf Gia sii t;„ = x f + xf'\i do: » «n = „ - l/ 3= +( v^/ 5j \ ' " " + (/ 3- -2v ^/ 5j\ ' " " ' - l V n = l , Lifu y Phep dat t;„ = u„ + (n = 1,2, ), d\X0c tim nhit sau: Xet h ^ so fix) = x^ + 3x2 _ i^hi u„+i = /(u„), Vn = 1,2, Ta c6 /(x) la da thiic b§c va fix) = 3x2 ^ ^//(^) = 6x + = ^ X = - Vay diem uon cua thi ciia ham so fix) Ik Ai-l, -1) Ta biet ring d6 thj ham so fix) nh$n diem uon ^ ( - , -1) lam tam doi xiing Do ta thUcJng doi he true toa d6 theo cong thiie doi true sau: | Y = y + l ^ toan 19 ta phai d^it t;„ = u„ + (n = 1,2, ) BM to4n 20 (De nghi thi OLYMPIC 30/04/2004) Cho day s6 (u„) nhu sau: ui = «„+i = 24u3 - 12V6u2 + i5u„ - ( „ = 1,2, ) Tin cong thiic so hg.ng tong qudt Un cua day so da cho 'jl , r V f>{x) = a„ > a„+i > • • • > nen a = Vi > a2 > hm a „ < o i < D o dc = n-«+(X) c h g n a—\ \%y hm a „ = T u u n g t U t a c h i i n g n i i n h dUdc: Tii n—•+0O 61 < < • • • < „ < „ + i < • • • < ! , Xo > X2 > l i m Xn = v/3 S U V r a h m „ = B a i t o a n 0 Cho U „ = 1, B a i t o a n 9 Cho l i m U „ + i = D o d o n—>+oo day so (a;„) nhu sau: XQ = 1, x „ + i = (1) l i m x „ = n—+00 x„ x,,Vn= 1,2, *:=1 +oo ^ •5n = //di A,7ii n —> -roo (/iz day so da cho c6 gidi han hHu han khong G i a i G i a s i i tt o n t i ^ i g i d i h a n h C u h a n • • • > II h m u „ = n—>+oo X2„+2 > h m X „ = v/3, d i e u n a y m a u t h u a n v d i ( ) V a y Xct day so (5„),"^rj nhu sau: hn < U2n < an, b„ < U „ + l < a „ ( n = , , )• hm X2„ > { x „ } la day so gidm, x „ > , V n e N * v d C u n g b k n g p h u d n g phap q u y n a p t a chiing m i n h dudc: n—>+oo •••> (lay SO d a c h o k h o n g h o i l u n—'+00 Suy r a X4 > S2n-v2 - S-m = x'2,,+1 - X2„4.2 > (do day ( x „ ) giam) l a CO phuong trhih: • ' ' , , „ • 27J+1 ^2' < 52n+i = X x^ 288 ^ ( - ) * + ' Xk = X i - ( X - Xa) ( X „ - X2„+l) < Xj Vay d a y {S2n}n=i ^5 ^ ' ^ " l i m S2„+i = l i m „^+oo n^+oo iS2n ^''^^ + X2n+i) = f i r d u g ti.f t a clu'rug m i u h ditcJc y = f{x) ''^'^ l i m 52„ + l i m X2„+i n-^+co n-.+oo l i m 52„ n^+oo = D o day so {5„}+ri h o i t u Chu y Tt/c/nif ta thu duoc vd Xn > 0, V n e N * vd kit qua sau: Cho { x „ } ; ^ ^ i Id day s6 l i m x „ = Xet day s6 {Sn)'^=i n—»+oo n s6 fix) y = (2)^ Y'^ — day so dd cho c6 gidi ,Vn=l,2, han hiiu han vd tlm gidi G i a i D e t l i a y x „ > , V n e N * X e t h a m so / ( x ) = X X = ^x hm han X2„-i = l i m X „ = ^ => l i m x „ = - 3 LiAi y H o (2) c o n dirdc g i a i n h u sa>i: D f i ( x ; y) l a n g h i e m cvia h e t h i x > j , V x > K h i = / ( a : „ ) , V n = 1,2, K h o n g m a t t i u h t6ng q u a t , g i a s i i 2/ > x D a t y = tx, k h i d o i > Thay vao ( a ) d u o c tx ^, 1 , < a < — ( d o i > 1) 27 y ' - 27 = a^, vdi a = H a m / u g h i c h b i e u t r e u [0; + o o ) , d o d o / ( x ) < /(O) = l , V x e [0; +cx)) V a y < x „ < l , V n = 1,2, Ta C O (1) ^ X i = 0, X = 1, X = — < f{X2n+2) =^ X2n+l Vay t h e o u g u y e n l y q u y n a p t o a n h o c suy r a day so {x2n-i}t=i < X2n+3' "^'^y ^o ^.aiig {x2„},tri l a day so g i a m D o d o ket lu.'l' v a {x2„},tri h o i t u (c6 gi6i h a n l i Q u h a n ) T u d n g t i r t a c l u i n g m i n h difdc d a y so vdi (1) s u y r a d a y so { x „ - i } ; ^ ^ i l i m X „ - i = y, n—•+00 vi /(x) lien tuc uen DhX, x = l i m X „ = x K h i d o t h e o (1) s u y r a < x , y < v a n-»+oo lim X2„= n->+c)o / fl\^V V27; '^^ ( x ; ^) = ^ / \ " (0 ^ < = ^ J o h{t) = Q ?i—+00 h m X2n-i] yn->+oo 290 = fiv)- j Iighich b i l u l a n g h i ? m d u y n h a t c i i a h§ t o a n 2 Cho day s6 { x „ } + f j nhu sau: xi = 0, x „ + i = l i m f{x2n-i)=f{ Thay ^ = - vao (1) d u d c V i / u g h i c h b i u u e u X4 = / ( X ) < / ( x i ) = X G i a s i i X „ - i < x „ + i , k h i do, f{X2n) Tren (0; + 0 ) , x e t h a i h a m so / i ( x ) - x , / ( x ) = a^ D e t h a y h a m / i d o n g hiea TOu h a m / u g h i c h b i e n , d o d o (c) c6 k h o n g q u a m o t n g h i e m X I < X3 vi / ughich bien neu ^2n > X2n+2 = y = - [27) Taco va y>0 !{X2n+\) (ca h a i h a m v a ( x ) c i m g u g h i c h b i e u t r e n k h o a n g (0; + 0 ) ) , d o d o d o t h i c i i a {xnj^.Z ( /(a'2u-l) > j y = x Bdi v a y ( x , y) l a n g h i e m c i i a (2) k h i v a c h i k h i x = y, n g h i a l a ( rdng (2) y = logjL X hai h a m n a y d o i x i i n g n h a u q u a d i r d n g p h a n giac c u a goc p h a n t u ' t h i i ; n h a t x i = 0, x „ + i = rninh ''[27, 4=> Ham so 5(2^) = l o g i X l a h a m ngiroc c i i a h a m / ( x ) = ( ^ Khi day so {Sn]t^i hoi tu Chifng fix) gidrn fc=i t o a n Cho day so 2/ = nhu sau: „ = 5]l(-l)'=Xfe,Vn= 1,2, Bai V a y t a c6 h e ,Vn= '^^S mtn/i Tdng ddy so dd cho cd gidi J 291 1,2, han hHu han vd tlm gidi han H U d i i g d a n T i t o i i g t i t hai toan l,^On hcji t u D a t ^ ^ h i r i ^ x , , , = a vh Jhn^X2n+i B a i t o a n ( D e t h i H S G q n o c gia, n i n i hoc 1999-2000, b a n g B ) tItUr (• > Day ,so { x „ } , s/c, = nni cdc hicu dnih diMc n = 0,1,2, = Xn+i \UrWc tht'tc dudi can la kh6ng,am viii irioi iiid Ui 7t vd ton G i a i D(' ( l u h i g n i i n l i d a y tai gidi + x^i ray (hmg the.o each Cho s6 sau: C + Xt) Chiing niinh han hHu han r&ng day { x „ } duoc => = ^ {jl'l' < c + XI, Tiroiig ti.r r a c l u i n g m i n l i d i M c b = s/c + xo < \f2c < c ffV) 16 (c S u y l a X i + i d\r(.lc xac chnh Vay t h c o nguyc-n l y q i i y n a p sny r a x „ ditrtc x a r 0 l a a _ ^ V i i y l i m X2n = l i m x2„+i, d o d o d a y so d a cho hen t u hcjp v d i ( « ) suy r a t r o n g m o i t r i t d n g h d p h a i d a y so ( x „ ) ^ ^ v a ( x n + i ) ' i * ' ' 292 D" l i a y -'•2A+2 < 2-2fc+4- Vay t h e o n g u y e n l y q u y n a p suy day so ( j ' u ) , t ^ ) Iiv day t a n g Tifclng t\t t a c lui^ng m i n h dUrJc day so (X271+1 TrvTdng a CC3 /(O) ^ l)t^ G i a si't X2k < X'2k+'2f{-i:2k+\) Vay' h a i d a y c o n ke u h a u c i i a d a y ( J „ ) [0; >/(•] T a xct T r i / c J n g h d p : x^ < x-z Suy r a J{XQ) fi-i-l) [0; v/?] 293 B a i t o a n ( D e t h i H S G q u o c g i a , n a m 2000, b a n g A ) Cho c la so thicc Day so { x „ } , n = , , , , duoc xdy dUng duang X n + \ Vcneu cdc bicu thtic dudi each sau: cho n - > + 0 - t-a d i t d c a = v c - v/c + a \/c + x^ ( n = , , , ) can Id khong dm Tim tat CO, cdc gid tri cua c dc ua^ moi gin tri ban dan XQ € ( ; c ) day { a ; „ } diMc ton tai gidi han hHu han Ihco xdc dinh (a'' + + - cn^) - (a^ + Tit < " < v^ t a + CO + c - c^) = c(c - 2) > 0"^"^° CO - « - c < 0, suy r a + o + - c = G i a i e a c h k h a c X e t h a m so / ( x ) = s/c - \ / c T x , V x e [0; c>2 N g i t o c l a i , b a n g p h i l d n g p h a p q u y n a p t h e o n , t a se d n i n g m i n h r K n g neu ( c - v/F+¥)' / (J^) = vcji ^ / T a CO - £ = 2vc-v/c + x c > t i l l Xn d U d c x a c d i n h v i m o i n = , , T h a t v§,y, d o X Q < c v a c > < 0, V x € , 0; 4v^c + x \ / c - v/c + X T i t c > 2, t a ( d C + X() < 2c =^ \/c + XQ < \/2c < C => C - \/c + XQ > S u y r a x i d i t d c x a c d i n h G i a s i i Xfc (fc = , , ) d a d U d c x a c d i n h K h i < + a - c a ) - (ca' ' 0, ( v ^ ) < Do d o p h i t d n g t r i n h / ( x ) = x c d n g h i e m d u y i i l i a t ae[0;^ Lagrange, v d i n d u I d n t o n t a i A;„ G ( x „ ; a) c h o |x„_2 - a| < • • • < fiXn) |xo - a| - f{a) = / ' ( f c „ ) ( x „ - tt) l a d a y s6 x „ + j = / ( x „ ) t h o a m a n suy r a l i m x „ = o, v d i a = L i f u y P l u t d i i g t r i n h x ^ + x + = c dUdc t i m r a n h u s a u : G i a s i i ( v d i < a < ^/c) k h i d o t i t Xn+i < |xn+i - A| < ^ l i m Xn ^«tJ6 = A/C - \ / c + x ^ ( n = 294 0,1,2, ), lim n-.+oo x „ = a |x„ - a| < • • • < Theo dinh ly Bai sau: t o a n (De nghi t h i O L Y M P I C 30/04/2001) xi Chitng mmh = - , xn+i = \xl Cho day so ( x „ ) nhu + x „ - (Vn = , , ) ( - h i J y a n rfpt /m?/ / i e n he bdi toan (