48 î £ | SO 8 2011 DauvoHpc DiKyyk HOC SANG TAO Ren luyen kT nang trong giai toan cho hpc sinh thong qua cac bai toan tmh tdng cua cac day so'''' I DAT VAN DE Toan hoc la mpt mon hoc ca ban trong chuang[.]
48 i^ - £ | SO 8-2011 DiKyyk HOC DauvoHpc SANG TAO Ren luyen kT nang giai toan cho hpc sinh thong qua cac bai toan tmh tdng cua cac day so' I DAT VAN DE Toan hoc la mpt mon hoc ca ban chuang trinh thong, la nen mong cho nhieu nganh hpc khac Hpc toan hay giai toan la yeu cau thuang xuyen mpi hoat dpng va suy nghT De co the hpc tot mon toan hay giai dupe cac bai toan doi hoi ta phai co cai nhin he thong ve mat kien thiic va ap dung viec phan loai cac dang kien thuc va ap dung vao cac bai toan vai lai giai dep tranh nhung sai lam, thieu sot Han niia de van dung giai dupe nhanh ta phai huy dpng dupe cac kien thiic lien quan den dang bai toan Tu khong nhirng gitip ta giai quyet tot bai toan ma ta co the phat trien sang tao them cac bai toan mai ma phuang phap giai hoan toan tuang tu Do la cong cu sac ben nhat cho nguai hpc toan La nguai nhieu nam true tiep giang day mon toan THCS, toi luon tran tra lam the nao de cho hpc sinh mam dupe cac phuang phap giai toan co he thong va van dung sac ben cac kien thiic vao giai cac bai toan de co ket qua cao cac ki thi Bai vay toi luon thay dupe trach nhiem cua minh la khong ngimg hpc hoi, tim toi a dong nghiep, tai lieu tham khao de huang dan hpc sinh bilt huy dpng nhanh kien thiic lien quan nhan dang bai toan giai dupe nhanh nhit, hpp ly nhat Khi giang toi thiet nghi lam each nao de hpc sinh nam dugc phuang phap giai t6i uu nhat, gpn nhe nhdt, la dieu toi hSng suy nghT nheu qua trinh giang day Toi nghT den viec " Ren luyen kl nang giai toan cho hpc sinh thong qua cac bai toan BUI HOANG ANH TrQdng THCS Cam Hda, Cam Xuyen, Ha TTnh tinh tong cua cac day so" la mpt each lam hay giiip hpc sinh nam vung co he thong canh giai cac bai toan nhanh nhat ma cac em liing tiing giai cac bai toan II NOI DUNG Nam khuon kho ciia van de neu tren pham vi bai viet toi chi xin neu mpt so kien thiic ca ban co su dung cac bai toan cu the cua phan ap dung giai toan, nhung kien thiic neu a day chi giai han nhiing dinh nghia, nhiing tinh chat ca ban quan trpng, khong de cap doi voi nhiing kien thiic ly thuyet ti mi va cac bai tap ap dung cho tung phan Trong qua trinh day hpc sinh khoi 6, toi nhan thay CO nhieu bai toan tinh tong, lien quan den tinh tong cua day so ma hpc sinh rat liing tiing, lam mot each may moc, rap khuon, thilu su sang tao Hau het cac em chi biet sii dung hau het cac phep tinh de tinh ket qua viec giai cac bai toan tren, nhung chua phan tich nim cac dang de tinh, chua biet dupe dac trung ciia tung bai Tir thuc te toi da chpn mpt so bai toan cu thi quen thupc ve day so dl di den nhung bai toan tong quat phuc vu giang day eho hpc sinh, nhiing dang khong chi giup cho hpc sinh tiep thu nhanh va giao vien ciing co mpt he thong kien thiic De nim ro vin de neu tren toi xin di vao mpt so bai toan cu the Vidul; Bai 1: Tinht6ng: Si = l +2 + + 9 + 100 43 DaijvaHcc soa^ZOVi Cdch sidi 1: Cap hai so dau va cuoi, cung nhu tung cap hai so each deu so dau va s6 cu6i diu co ting 101 S] CO 50 cap nhu the, do kit qua: S, = 101 50 = 5050 Cdch gidi 2: + + + + 99 Nen: + + 101 + Si= 100 Si = + Si = 101 + + 100 + 99 101 + 101 (co 100 hang so) 101.100 Do do: Si = = 5050 Tii hai each giai 1,2 a SGK, cac em lam dupc bai tong quat sau: Bai 2: Tfnh tong: S = l + + + (n l) + n De dang tinh dupe: _ in + \).n Bai tap ren luyen: Ti= + + + 2n T2= + + + (2n+l) Tu cac bai tap tren ta co bai tinh tong tong quat cua mpt day cac so each deu Bai 3: Tinh ting: S3 = + a2 + + an-I + an vai a2 = a3 = = Sii dung each giai ta tfnh dupc: _ (fl|+a2)-» ^'^ Vay tinh mpt day co dang each diu ta phai xac dinh s6 hang diu va s6 cu6i tim so hang ciia day Bai 4: Tim tong cua cac s6 tu nhien chia hit cho du va nho han 2003 Giai: S6 hang diu la va so hang culi la 2001 2001-1 so so hang la: • -h = 505 so hang Den day hpc sinh de danh tfnh dupc ting nha ap dung cong thiic tren, hoac hpc sinh ap dung hai each giai va cflng tfnh dupc Neu bai toan tren chi dung lai a hai each giai da trinh bay thi chua dii ma phai tim toi, phat hien each giai khac niia Cdch gidi 3: Co the hpc sinh du doan sau dung phuang phap quy nap de chiing minh cac cong thiic tong quat Cdch gidi 4: Bai va bai khong phan tfeh giai nua, ta giai bai tap tong quat bang phuang phap khii lien tiep Cac so hang chi eo mpt thua so moi so lien tiep each K dan vi Vay ta nhan hai ve vai 2K Ta co: a2 - = as - a2 = = an - ap -1 = K Ton tai so T, M cho a2 T = 33 - = = an -an.2 =M-an-i Nhan hai ve vai 2K ta dupc: 2K S3 = K a i + K a + + K an.i + 2K an 2K S3 = (a2 - T) +(a3 - a,) a2 + + (anan-2).an.i ( M - a n - l ) An 2K S3 = a2 - T + a2 a3 a2 +• + an-i an - an-2 an-1 + an M - an-1 an 2K.S3 = an.M T.ai a M -T.a, Cong thiic chinh la 2K cong thiic a bai Tu each giai tren diu co mpt muc dfch la de tfnh tong cac day nhu neu a tren, rieng each giai CO mpt muc dfch lan han nua dl giai cac bai toan khong theo quy luat tren 83 = Vi du 2; Bai 1: Tfnh ting S4 = + + n (n 1) 50 SO 8-2011 DaijvaHpc — Neu nhu giai bai toan theo each 1, a vf du thi khong bao giof giai dupe nhung phan tfeh each giai thi viec giai bai toan rat don gian vf du ta nhan ve vai 2K nhung a bai ta nhan ve vai 3K truang hop bai toan vai K = Tac6:3S4 =1.2.3 +2.3.3+ + n(n+l).3 3S4 = 1.2.3 + 2.3.(4 1) + + n(n + 1) {(n + 2)-(n 1)} 3S4 = 1.2.3 + 2.3.4 2)-(n l ) ( n + l ) /SGAY SAY Bai 4: Tinh tong: S7 = aia2a3 + a2a3a4 + + an an +1 an+2 Vaia2-ai = 33-32= Ton tai M va T cho = an+2-an+i =K a4 T = a5 = = an+2 - an.2 = - M - a„.i = 4K Bang each nhan ve voi 4K, dung cac phep biln doi nhu tren ta tfnh dupc: 1.2.3 + + n(n + l)(n + 4K ^7- 3S4 = n(n+l)(n + 2) n(n + l)(n + 2) S4= - - Tuang tu ta tfnh dupc tong cua day cac s6 hang CO nhieu thira so theo quy luat tren bang each tuang tu Tii bai toan hudng dan hpe sinh di tim bai Vi du 3: toan tong quat han Bai 1: Tfnh tong Bai 2: Tfnh tong + S5 = aia2 + a2a3 + + anan+i 1.2 2.3 n(n + l) Vai a2- = a3 - a2 = = an+i - an O bac Tieu hpe da giai dupc: Ton tai hai so T, M cho 1 1 -f a3 - T = a4 = = an+i an-2 = M an-i = n n+1 3K Ta nhan hai ve vai 3K: Ss-l ^«+l 3K S5 = 3Kaia2 + 3Ka2a3 + + 3Kanan +1 Bai 2: Tinh tong 3K.S3 = (a3 T)aia2 + (a4 - ai)a2a3 + + (M an-i)anan+i + 3K.S5 = aia2a3 aia2 + a2a3a4 aia2a3 + + ' 1.3 3.5 (2n + l)(2n + 3) anan+iM-an-i.an+i Ta biet a bai hai thira s6 a mlu each 3K.S5 = an.an+iM Taia2 mpt dan vi bai hai thua s6 a ciing mot mlu each hai don vi Vay ta nhan hai vl v6i c _ an-an^,.M T.a,.a2 2: 3K Bai 3: Tfnh tong 2SQ = — + — + + (2n + l)(2n + 3) 1.3 3.5 S6 = 1.2.3 + 2.3.3 + + n.(n+l).(n+2) 1 1 Tuang tu nhu bai tap ta nhan hai ve vai 4, de + • 2S9=1 - + + dang bien doi dupc 2« + l 2« + s,= ±.± n.(n + l).(n + 2).(n + 3) S6 = 4K Tuang tu ta giai bai tong quat SQ = 12« + 3, Tu bai toan tren ta co bai toan tong quat sau: DaUvaHpc so 8-2Q^^ m Bai 3: Tinh tong doi lupng nhan vao a vf du ta biln dii cac so da cho 1 + +, a^a2 ô2ô3 Sio + ã Ta xet sang mpt loai toan mai: ««««+! V a i a - a i = a3-a2= Vi du 4: = an+i = K Dung phuang phap giai nhu bai 1, bai ta eo: >io 1 a, V."i :K