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TRAN VINH NHA XUAT BAN HA NOI 4t_ IRAN VINH THIET KE BAI GIANG DAI SO VA GIAI TICH ir''-/ ".V.r' •» It: isiiirGCiAo TAP HAI NHAXUATBANHANOI THIET KE BAI GIANG DAI sd VA GIAI TICH 11 - NANG CAO - TAP HAI TRAN VINH NHA XUAT BAN HA NOI Chiu trdch nhiim xudt bdn : NGUYEN KHAC OANH Biin tap : PHAM QUOC TUAN ViBia TAO THANH HUYEN Trinh bdy : QUYNH TRANG Sica bdn in : PHAM QUOC TUAN In 1000 cuon, tai Xf nghiep In ACS Viet Nann. Km 10 - Dudng Pham Van Dong - Kien Thuy - [Hai Phong. Giay phep xuat ban so: 208 -2007/CXB/46 m TK - 47/HN. In xong va nop luu chieu qu^ I nam 2008. Cki/ONq III DAY SO. CAP SO CONG VA CAP SO NHAIN Phan 1 ^mUrn^ VAX D^ CUA CHUWSTG I. NOI DUNG Noi dung chinh cua chuung III: Phuong phap quy nap toan hoc: Dinh nghia, cac bu6c chiing minh bang phuong phap quy nap. Day so: Dinh nghla day so la gi ? Day sd hihi han va day s6' yo han, cong thiic tdng quat ciia day sd, cac phuong phap cho day sd, day sd tang, day sd giam va day sd hi chan. • Qip sd cdng : Dinh nghia, cdng sai, sd hang tdng qiiat, tinh ch&'t cac sd hang, tdng n sd hang dSu tien cua cap sd cdng. • Cap sd nhan : Dinh nghTa, cdng bdi, sd hang tdng quat, tinh chat cac sd hang, tdng n sd hang dau tien ciia ca'p sd nhan. II. MUC TIEU 1. Kien thiirc Nam duoc toan bd Icien thiic co ban trong chuong da neu tren, cu the : - Biet chiing minh va nhan bie't khi nao sit dung phuang phap quy nap toan hoc. Bidt tim cac sd hang tdng quat cua day sd; Chiing minh duoc day sd la day sd tang, giam, day sd bi chan. • Nam duoc Ichai niem va each nhan bie't mot day sd la ca'p sd cdng ; tim duoc sd hang tdng quat va tinh tdng n sd hang dau tien cua mot ca'p sd cdng. Nam duoc khai niem va each nhan bidt mot day sd la ca'p sd nhan ; tim duoc sd hang tdng quat va tmh tdng n sd hang dSu tien cua mot ca'p sd nhan. 2. Kinang Vun dung cac budc quy nap de chiing minh bai toan bang phuong phap quy nap toan hoc. van dung thanh thao c^c tinh chat cua ca'p sd cdng va cap sd nhan trong giai toan. Bie't each cho mot day sd, each khao sat tmh tang, giam ciia cac day sd dcfn gian. Nhan bie't duoc cap sd cdng, cap sd nhan ; bie't each tim so hang tdng quat va tdng n sd hang d^u tien ciia cac ca'p sd do trong cac trudng hop khdng phiic tap ; Bie't van dung nhiing kie'n thiic trong chuong de giai quye't cac bai toan cd lien quan dugc dat ra d cac mdn hoc khac, cung nhu trong thuc tiln cude sd'ng. 3. Thai do • Tu giac, tfch cue, ddc lap va chii ddng phat hien cung nhu ITnh hdi kie'n thUc trong qua trinh hoat ddng. • Can than chinh xac trong lap liian va tinh toan. Cam nhan dugc thuc teciia toan hoc, nha't la dd'i vdi day sd. Pban 2 CAC BAI SOA^ §1. Phu'cfng^ phap quy nap toan hoc (tie't 1, 2) I. MUC TIEU 1. Kien thtic HS nam dugc : • Phuong phap va cac budc chiing minh quy nap. Khi nao thi van dung phuong phap quy nap. Giai thich dugc phuong phap quy nap. 2. KT nang Van dung thanh thao phuong phap quy nap trong giai toan. Bie't them mdt phuong phap chiing minh dd'i vdi bai toan cd lien quan den sd tu nhien. 3. Thai dp Tu giac, tich cue trong hgC tap. « Biet phan biet rd cac Idiai niem co ban va van dung trong timg trudng hop cu the. . - Tu duy cac va'n de ciia toan hgc mdl each logic va he thd'ng. II. CHUAN BI CUA GVVA HS 1. Chuan bj ciia GV Chuan bi cac cau hdi ggi nid. Chuan bi cdc vi du sinh dgng. • Chudn bi pha'n mau, va mdt sd dd dung khac. 2. Chuan bi ciia HS - Can dn lai mgt sd kie'n thiic da hgc ve sd tu nhien d ldp dudi. IIL PHAN PHOI TH6I LUONG Bai nay chia lam 2 tie't : Tii't 1 : Tix ddu din hit vi du 1. Tii't 2 : Tiep theo den het phdn bdi tap. IV- TIEN TRINH DAY - HOC A. DAT VANDE Cau hdi 1 Xet tmh diing - sal cua cac cau sau day : a)Ne'ua>bthia" >'b" b)Ne'ua>b> 1 thia" > b", Cau hoi 2 Cho cac menh 6i sau : a) Sd nguyen duong le ldn hon 1 la so nguyen to. b) 1 + 2 + 3 +, + n = "^""^^\ n e N. 2 Hay xem xet tinh diing sai cua cac menh de tren vdi 3 sd hang dSu tien. B. BAIMdl , HOATDONCl 1. Phuong phap quy nap toan hoc « GV dat va'n de : HI. Hay phat bi^u mdt vai menh de chiia bie'n tu nhien A(n). • Sau dd GV neu bai toan trong SGK. Chdng minh rang ydi mgi so nguyen duang n, ta ludn cd 1.2 + 2.3 + t n(n + 1) = x : • Thuc hien [HI] trong 4' Hoat dong cua GV Cau hdi I Hay kiem chiing khi n = 1. Cau hoi 2 Cd thd kiem tra dang thiJc (1) vdi mgi n dugc khdng? Hoat dong cua HS Ggi y tra Idi cau hoi 1 Vdi n = 1 ta cd dang thiic luon ludn diing. Ggiy tra ldi cau hoi 2 Khdng thd. • GV neU cac budc quy nap ; • Bu&c 1 (budc ca sd, hay budc khdi ddu). Chieng minh A(n) la mpt menh de diing khi n = 1. • Bu&c 2 (budc quy ngp, hay budc "di truyen"). Vdi k Id mpt sd nguyen duang tuy y, xudt phdt td gid thii't A(n) Id mpt menh de diing khi n = k, chimg minh A(n) cdng Id mpt minh de diing khi n = k + 1. Ngudi ta ggi phuang phdp chimg minh viCa niu tren Id phuang phdp quy ngp todn hgc (hay cdn ggi tdt Id phuang phdp quy ngp). Gid thie't dugc noi tai a budc 2 ggi Id gid thie't quy ngp. • GV dua ra mdt sd cau hdi ciing cd : H2. Hay giai thich tai saO phep chiing minh bang quy nap la diing. H3. Phep chiing minh bang quy nap cd ap dung cho bai toan chiing minh menh de A(x)ba'tki. HOATDONC 2 2. Vi du ap dung • Thuc hien vi du 1 trong 5 phiit. Hoat ddng cua GV Cau hoi 1 Xet tinh dung sai ciia cong thiic vdi n = 1. Hoat dong cua HS Ggi y tra Idi cau hoi 1 Ta tha'y n = 1, cdng thiic tren ludn ludn diing. Cau hoi 2 Gia sii cdng thiic diing n = k hay thie't lap cdng thiic. Cau hoi 3 Hay thie't lap cdng thiic khi n = k + 1 va chiing minh cdng thiic do. Ggi y tra Idi cau hoi 2 4 Ggi y tra Idi cau hoi 3 l^ + 2^ + + k^ + ik+l)^ = (k + 1)2 (it + if 4 ' ; HS tu chiing minh tie'p. Thuc hien 1H2J trong 5'. Hoat dgng cua GV Cau hoi 1 Xet tfnh diing sai ciia cdng thiic vdi n = 1. Cau hoi 2 Gia sii cdng thiic diing n = k hay thie't lap cdng thiic. Cau hoi 3 Hay thie't lap cdng thiic khi n = k + 1 va Chiing minh cdng thiic do. Hoat dgng cua HS Ggi y tra Idi cau hoi 1 Ta tha'y n = 1, cdng thiic tren ludn ludn diing. Ggi y tra Idi cau hoi 2 HS tu thie't lap. Ggi y tra Idi cau hoi 3 l+3+ + (2)t-l) + (2^+l) = (^+lf. HS tu chiing minh. • Thuc hien [H3J trong 5' Hoat dgng cua GV Cau hoi 1 Xdt tihh diing sai cua cdng thiic vdi n = 1. Hoat dong cua HS Goi y tra ldi cau hoi 1 2 Vdin= 1, tacd 1 =1 1(4.1'-1) 3 [...]... cdng ihUc vdi n = 1 9 ^ 2 = 4 = '' Cau hdi 2 2.1(1 + 1X2.1 + 1) , „ ^ '^ ^Nhuvay, 3 dung khi « = 1 Ggi y tra Idi cau hdi 2 Gia sii cdng thiic diing n = k Cau nay GV chi neu van de, HS tu lap hay thie't lap cdng thiic theo k Cau hdi 3 Ggi y tra ldi cau hdi 3 Hay thidi lap va chiing minh Ta se chiing minh cdng thUc vdi n = k + 1 2^ + A^ + + {2kf + {2k + 2f, 2{ k+\)(k + 2) {2k + 2, ) 3 Bang phuong pha'p... cdng Vdi « = 2, ta cd thiic vdi n = 2 1 _ 3 _ 2 +1 4 4 2. 2 Nhu vay, (1) diing khi h - 2 C a u hdi 2 Gia SU" cdng thiic dung n = k hSy thiet lap cdng 14 Ggi y t r a ldi cau hdi 2 cau nay GV chi ndu va'n de, HS tu lap thuc theo k Cau hdi 3 Hay thie't lap va chiing minh cong thUc vdi n = k + 1 Ggi y tra ldi cau hdi 3 ta cd / 1 1 k+l 2k 1— 9 1 1 k'J (k + l)' j kik + 2) , k + 2 (/t + l )2 2(k + \) Tii caC... thUc vdi n = k + 1 1 1 k + 2 k + 3 '" 1 2k + \ 1 13 2{ k + \) 24 Bang cac phan tfch : 15 1 1 -; + + k +2 k+3 ^ 1 1 1 2k ' 2k + l 2ik + l) 1 = 1 + ^+1 k +2 2A: + 1 1 + + — 2k 2( it + l) k+l HS tur chiing minh tiep Bai 6 Sii dung phuong phap quy nap Hoat ddng cua G V Cau hdi 1 Xet tfnh diing sai cua cdng thuc vdi n = 1 Hoat ddng cua HS Ggi y tra ldi cau hdi 1 Vdi « = 1, ta cd Hj = 7 .2^ ' ~^ + 3^-' ~ ' = 7 +... 2 Hay chgn khang dinh diing Ggi y tra ldi cau hdi 2 Cac khang dinh diing : b), c), d) va e) • Mdt sd cau hdi on tap H 22 Day so la mdt ham so (a) Diing ; (b) Sai H23 Mgi day sd deu la day sd tang hac giam (a) Diing ; 26 (b) Sai H24 Day sdbi chart la day sd tang,hac giam (a) Diing; < (b) Sai H25 Day sd giam la day sd bj chan (b) Sai (a) Diing ; H26 Day so tang la day so bj chan (a) Diing; (b) Sai H27... \) Tii caC chiing minh tren suy ra (1) diing vdi mgi so nguyen « > 2 Bai 5 Sii dung phuong phap quy nap Hoat ddng ciia GV Cau hdi 1 Xet tfnh diing sai cua cdng thiic vdi n = 2 Hoat ddng cua HS Ggi y tra Idi cau hdi 1 Vdi rt = 2, ta cd 1 3 1 _ 7 _ 14 13 4 "^ 12 " 24 ^ 24 • Nhu vay, (1) diing khi « = 2 Cau hdi 2 Gdi y tra Idi cau hdi 2 Gia sii cdng thiic diing n = k cau nay GV chi neu van de, HS tu lap... day so nay • GV neu day sd : •'-" """ 3«+i • : 21 Thuc hien iH2| trong 5 Hoat ddiig cuai GV Hoat ddng cua HS Cau hoi 1 Ggi y tra ldi cau hdi 1 Hay xac dinh sd hang U33 cua HS tu tinh toan day so trdn r,A A8 Dap so Ux\ = — a 25 Ggi y tra ldi cau hdi 2 Cau hdi 2 Hay xac dinh sd hang U333 cua day so tren HS tu tfnh toan r,A 83 Dapso "333 =22 5" • GV ndu each 2: Cho ddy sdbdi hi thitc truy hoi (hay cdn ndi:... cd 1 < 2 V l Nhu vay, ba't dang thiic diing khi « = 1 13 Cau hdi 2 Ggi y tra Idi cau hdi 2 Gia sir cdng thiic diing n = k Cau nay GV chi ndu va'n de, HS tu lap hay thie't lap cdng thiic theo k Cau hdi 3 Ggi y tra Idi cau hdi 3 Hay thie't lap va chiing minh cdng thUc vdi n = k + 1 1 + — '" N /2 l^k . minh 11 1 1 13 k + 2 k + 3 '" 2k + 2{ k + ) 24 Bang cac phan tfch : 15 1 1 -; + + k +2 k+3 ^1 1 1 2k ' 2k + l 2ik + l) 1 1 1 = + + + — ^+1 k +2 2k 2A: . HS tu lap. Ggi y tra ldi cau hdi 3 Ta se chiing minh 2^ + A^ + + {2kf + {2k + 2f, 2{ k+)(k + 2) {2k + 2, ) 3 Bang phuong pha'p quy nap. HS tu chiing minh. Bai 3. Ihcdng . k'J 1 (k + l)' j k + l kik + 2) , k + 2 2k (/t + l )2 2(k + ) Tii caC chiing minh tren suy ra (1) diing vdi mgi so nguyen « > 2. Bai 5. Sii dung phuong phap quy nap