5.1. Tong Fibonacci c6 trong so'
d dinh ly 1, ta da chiJng minh du-dc cong thiJc Fi = F„+2 - 1.
i=l
Tuf day, mot bai toan mdi du^dc dat ra la hay tinh tong Yl vdi Oi la cdc so" nguyen du'dng. i = i
Tru-dc het ehiing ta bat dau bang bai todn ddn gian nhat cua tong tren v d i a, z d bai toan sau:
Bai toan 32. Vdi moi so nguyen duang n, tinh tong sau:
Fi + 2F2 + • • • + nF„.
Lori giai. Dat An = Fx+F2 + --- + Fn vk Bn = F i + 2F2 + - • • + nF„.
Ta CO
5 „ = F i + 2F2 + 3F3 + • • • + nF„
h n n n n
1=1 i=2 i=3 i=n ' •
= An + (An - A,) + (A„ - ^ 2 ) + • • • + {An - An-l) n- 1 n- 1
= nAn-^A = n{Fn+2 - 1) - Y^{F,+2 - 1)
1=1 i=i ;
= nF„+2 - n - {Fn+3 - 3) + (n - 1) - nF„+2 - F „ + 3 + 2.
v a y Bn = nFn+2 - Fn+3 + 2. , . . . i •
124 Cdc phumg phdp gidi todn qua cdc ky thi Olympic Bai toan 33. Vdi moi so nguyen duang n, tinh long sou:
Cn = Yl {2i - 1)F2,-1.
i=i
Lcfi giai. De tinh tong Cn, chung ta se ap dung ket qua cua dinh ly 2:
" FS + • • • + F2„_l = F2n, Fo + F2 + • • • + F2„ = F2„+l - 1.
Dat F)„ = F i + Fa + • • • + F2„_i, ta CO
C„ = F i + 3F3 + 5F5 + • • • + (2n - l)F2„_i
• i = ^ F 2 . _ i + 2 ^ F 2 . _ i + 2 ^ F 2 , - i + --- + 2 ^ F 2 . _ i
i=l t = 2 t = 3
= D „ + 2(F>„ - D i ) + 2(Fằ„ - F>2) + • • • + 2(D„ - F>n_i)
n - l ằ-l
= D „ + 2(n - - 2 ^ A = (2n - l ) ! ) ^ - 2 ^ F2.
i=i i=i
•
= (2n - l)Dn - 2(F2„_i - 1) = (2n - 1)F2„ - 2(F2„-i - 1)
= ( 2 n - l ) F 2 „ - 2 F 2 „ _ i + 2 . Vay Cn = (2n - 1)F2„ - 2F2„_i + 2.
Bai toan 34. Vdi mdi so nguyen duang n, tinh tong sau:
^ ( 2 n -2i + l ) F 2 i _ i .
m giai. Dat En = ZUi^^ ' + De' tinh to'ng nay, ta 1=1
se siJ dung dinh ly 2 va ket qua bai toan tren. Ta c6
n
Cn + En = Y,{2i - l ) F 2 . - i + Y.{2n -2i + l)F2i-i
= [{2i - 1) + (2n - 2i + 1)] F2ô_i = 2nJ2 ^2^-l
<• a;.
i=l
= 2nF2„.
Xung quanh so Fibonacci 125
Do do:
F„ - 2nF2„ -Cn = 2nF2n - [{2n - 1)F2„ - 2F2 „_i + 2
- F2„ + 2F2„_i - 2 = (F2„ + F2„_i) + F2„_i - 2,
= F2„+l + F 2 n - 1 — 2.
vay En = F2n+i + F2n-i - 2. jj.;.- :^ rii'Vi ^ •
De ket lai phan nay, mdi ban doc ciing thu: siJc vdi bai todn tinh tong sau: . ^'A + if^.lf.'i^l \ -s* unJôằ -t- VIBW st+.i .'i**
Bai toan 35. V(^/ mo/ ^-o nguyen daang n, tinh cdc tong sau:
(a) E I L l ( 2^ ) F 2 ^ (Ddp ^d; 2ni^„+i - 2i^„.) ' '
(b) Er =i(2" - 2z + 2)F2,. 2F2„+2 - 2n - 2.)
5.2. So Fibonacci va bat dang thufc trong tarn giac
Day la mot tim toi mdi diTdc phat hien trong thdi gian gan day ve so Fibonacci. Do'i vdi hoc sinh chuyen Toan, ngoai kha nang giai todn tot thi kha nang tim toi va sang tao hay phat hien ra nhiJng dieu mdi cung rat quan trong. pj-
Bai toan 36. Cho tarn gidc ABC. Chiing mink rang ..
a'^Fn + 6 'F„+i + C2 F „+2 ^ 45x /F„F„+i + F„+iF„+2 + F„+2F„.
Chiang minli. Dat = VF„F„+i + Fn+iF„+2 + Fn+2Fn- Khi do, bat dang thiJc can chu'ng minh cd the dufdc viet lai thanh:
a^Fn + b^Fn+l + C ^Fn+2 > kS, hay
a^Fn + h'^Fn+i + (a^ + 6^ - 2a6cosC )F„+2 ^ hiahsinC.
Quy dong mau so' vd bang cac phep nhom ddn gian, ta viet du'dc bat dang thifc du'di dang:
2a ^(F„ + F„+2) + 262 (F„+i + F„+2) - a&(4F„+2 cos C + A: sin C) ^ 0.
126 Cdc phuang phdp gidi todn qua cdc ky thi Olympic
hay Wdng duWng
^ ( F „ + F„+2) + - ( i = ^ n + i + F „ + 2 ) - ( 4 F „ ^ 2 C o s C + f c s i n C ) ^ 0. (*)
b o
Tdi day, siJ dung bat dang thiJc Cauchy-Schwarz, ta c6
(4F„+2cosC + ksinCf ^ + k^){cos^C + s i n ' C ) ,
tilf d6 suy ra , ^ Siil
4Fn+2 cosC + A;sinC ^
= 4v/(F„ + Fn+2){Fn+r + (1) Mat kh^c, dp dung bat dang thufc A M - G M , ta lai c6 { (t-
^(F„ + F„+2) + - ( F „ + l + ^n+2)
^ A^{Fn + F„+2)(i^n+l + Ket hcJp (1) (2) lai, ta suy ra (*).
Bai toan 37. Cho tarn gidc ABC. ChvCng minh rang
(2)
•
/n+2
a'Fr. + b'Fn+, + c'Fr.^2 > ^ S [ ^ F ^ - Fl^,
\fc=l >
Chiirng minh. SuT dung ket qud bai toan tren, ta c6
-M^m = ^S^JFnFn+l + i ^ 2 -
Mat khdc, theo dinh ly 4 , ta lai c6
F2 + F2' + ... + F2 = F „ F „ + I . Do d6, ket hdp vdi danh gia d tren, ta thu du-dc ngay:
a^F^ + 6'F„^i + c'F„+2 ^ 4 5 /F2 + F ' + • • • + F^ + F'^^
= 45
/n+2 \
+ 1
Day chinh la ket qua can chuTng minh. •
Xung quanh so Fibonacci 127
Bai toan 38. Cho tarn gidc ABC. ChvCng minh rang " y ^ . i
^ (a^Fn+iFn+2 + b\/F„+2F„ + cVF„+2F„)^
Chtfng minh. Ket qud nay du'Oc suy ra triTc tiep tuf bat dang thiJc
Cauchy-Schwarz. •
5.3. So Fibonacci va ti 1^ vang
Ti le vang a, diTdc dinh nghia la ti so khi chia doan thang thanh hai phan sao cho ti le giffa ca doan ban dau vdi doan Idn hdn bang ti so giffa doan Idn va doan nho. C6 the chiJng minh rang neu quy do d^i doan Idn ve ddn vi thi ti le nay la nghiem dUdng cua phu'dng trinh x ' - X - 1 = 0.
"Ti le vang" diTdc cho 1^ hang so nit ra tuf day Fibonacci 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . - day v6 han cac so t\X nhien bat dau bang hai phan tu* 0 va 1, cac phan tuf sau do du:dc thiet lap theo quy tac moi phan tuf luon bang tong hai phan tu* tnrdc no. Dieu dang noi la, ti so' giila hai so lien tiep nhau cua day so do ngay cang tien den so ty le v^ng la 1.618. . ^
Day so Fibonacci cung nhif so' " t i le vang" la nhffng bi an Idn cua Toan hoc. Chung ton tai khap ndi, trong tiT nhien, trong hoi hoa, kien true, t ^ i chinh, khoa hoc... V i du nh\i, neu chia tong so' ong cdi cho tong so ong diTc trong mot to ong bat ky, se cd gii tri la 1.618. Hay tren cd the hau het nhCTng ngiTdi binh thiTdng: ti le cua chieu cao cd the chia cho chieu cao tuf that lifng trd xuo'ng hoac ti le chieu dai canh tay (tinh tiT vai den dau ngdn tay) chia cho chieu dai tuf khuyu tay tdi dau ngdn tay deu cho ket qua xap xi 1.618.
"Ti le vang" cung diTdc tim tha'y trong kie'n true cd nhu" Kim ty thap A i Cap, dien Parthenon d Hy Lap hay trong cac biJc t^dng CO va nhffng biJc tranh thdi ky Phuc hUng. No bieu hien cho quan niem ve cai dep va sU can xufng, u'a nhin.
128 Cdc phuang phdp guii Uuiit cjiia ccu ky llii Olympic Tam giac vang s\mm -"Ii '^Mh. "i^x'tv; i " " " ' , f,iV;'= J "
Mot tam gidc can c6 ti so giQa canh ben vd canh ddy bang ti so vdng duac goi Id tam gidc vdng.
•r,;'i
Chiing ta se tim hieu mot so tinh chat cua tam giac vang.
Bai toan 39. Cho tam gidc vdng ABC vdi canh ddy AC. D Id diem chia canh BC theo tl so vdng vd BD > CD. Chiing minh rang AD Id dudng phan gidc trong goc A.
Chu'ng minh. Do D chia canh BC theo ti so vang nen ta c6 , BD
B
Mat khac, do ABC la tam giac vang nen ta cung c6 AB
AC = a.
Tvi hai ket qua tren, ta thu difdc
: ' ' BD_AB CD ~ AC
Dieu nay chi^ng to AD la diTcJng phan giac trong goc A ciia tam
giac ABC. • Nhan xet. Ta cung chu'ng minh du'dc rang: Neu ABC Id tam
gidc vdng vdi canh ddy AC vd D Id chdn dUdng phan gidc trong goc A thi CAD cUng Id mgt tam gidc vdng.
Xung quanh so Fibonacci 129
Bai toan 40. Chiing minh rang neu tam gidc ABC Id tam gidc vdng thi goc d dlnh bang 36°. ^'ru i^Lt to ftem rtniyj •tL:, \'a/t
Chu'ng minh. Gia su" ABC la tam giac vang vdi AB = BC = aAC.
Goi D la diem chia BC theo ti so vang nhu" hinh ve bai 39. Dya vao nhan xet d tren, ta c6 CAD cung la mot tam giac vang va dong dang vdi tam giac ABC, suy ra
ACB = ADC = 2DAC.
Dat goc ACD = 2x thi tuf AACD, ta c6
2x + 2x + x= 180° =^x = 36°.
Vay ^ - C = 72°, 5 = 36°. • Sau day la mot so bai toan khac, mdi ban doc ciing thu: siJc.
Bai toan 41. Chiing minh rang neu tam gidc ABC can, canh ddy AC, CO goc d dinh bang 36° thi tam gidc ABC Id tam gidc vdng.
Bai toan 42. Cho tam gidc ABC can dlnh B. D Id mot diem tren canh BC sao cho AABC dong dang ACAD. Chiing minh rang
ABC Id tam gidc vdng khi vd chi khi = a.
Hinh chif nhat vang
Hinh chU nhdt c6 tl so chieu ddi chia cho chieu rong bang tl so vdng goi Id hinh chQ nhdt vdng.
A B
D
•I h.
c •'
130 Cdc phuang phdp gidi todn qua cdc ky thi Olympic Mot so hinh chu* nhat v^ng trong thiTc tien la cac bufc hoa noi tieng, hay cdc cong trinh c6 dai tren the gidi c6 ti le vang. Chang han nhif hinh tufcJng chiia Jesus, biJc tranh nang Mona Lisa cua dai danh hoa Leonardo Da Vinci:
Hinh chi? nhat vang cung xuat hien d dong ho, hay cong trinh kien true CO.
Xung quanh so Fibonacci 131
5.4. Day so' Fibonacci trong trf nhien va kinh te - xa hQi
Day so' Fibonacci trong trf nhi^n 5 , , Day Fibonacci xuat hien d khap ndi trong thien nhien. Nhffng chiec
\i tren mot nhanh cay moc each nhau nhifng khoang tu'dng iJng vdi day so Fibonacci. Cdc so Fibonacci xuat hien trong nhifng bong hoa. Ngoai ra, so Fibonacci con difdc tim thay trong su" sap xe'p choi la tren than cay, stog dong vat, gia pha loai ong difc, dang vong xoay trong qud thong, trai khom (thdm)... |: ,
Hau het cac b6ng hoa c6 so' cdnh hoa la mot trong cdc so' 3, 5, 8, 13, 21, 34, 55 hoac 89. Hoa loa ken c6 ba canh, hoa mao lUdng vang c6 5 canh, hoa phi ye'n thtfdng cd 8 canh, hoa cue van tho cd 13 canh, hoa cue tay cd 21 canh, hoa cue thu'dng cd 34 hoac 55 hoac 89 cdnh.
Cdc so Fibonacci cung xuat Men trong cdc bong hoa hiTdng du-dng. Nhu-ng nu nho se ket thanh hat d dau bong hoa hu-dng
132 Cdc phuang phdp gidi todn qua cdc ky thi Olympic difcJng diTdc xep th^nh hai tap cac difdng xoan 6'c: mot tap cuon theo chieu kim dong ho, con tap kia cuon ngu'cJc theo chieu kim
dong ho. So cac diTdng xoan oc hu'dng thuan chieu kim dong ho thudng la 34, c6n ngufdc chieu kim dong ho la 55. Doi khi cac sS' nay la 55 va 89, va tham chi la 89 va 144. Tat ca cac so nay deu la cic so Fibonacci ke tiep nhau (ti so cua chiing tien tdi ti so' vang).
Pln*appl* tnd it'i atmn
Trai thong ciing vay. Vong xoay tuf trung tarn c6 5 va 8 nhdnh.
Tr^i khdm c6 ba xoay la 5, 8 va 13. Lai mot bang chiJng nhiJng con so' nay khong phai ngau nhien. Thien nhien chdi tro Toan hoc vdi Chung ta?
Khong ai biet nhtfng cac khoa hoc gia suy doan rang cac loai thUc vat moc theo hinh the xoay oc theo nhifng con so' Fibonacci vi no tie't kiem nhieu be mat hdn. Sap xe'p nhu" the', chung gia tang dieu kien tang tn^dng va do do, nhieu dieu kien sinh ton hdn.
Day s6' Fibonacci trong Sm nhac
Ban vanxd Fibonacci la mot ban nhac ma giai dieu cua no bat nguon tiif mot trong nhifng day so' noi tieng nha't trong Ly thuyet so - day so Fibonacci. Hai so' dau tien cua day la so 1 va so' 2, cac so tiep theo difdc xac dinh bang tong cua hai so lien tiep ngay tru'dc no trong day.
Kung quanh so Fibonacci 133
Ban vanxd Fibonacci thu difdc bang viec chuyen day so Fi- bonacci thanh day cac no't nhac theo qui tac chuyen mot so nguyen du'dng thanh not nhac sau day: ^ , , ,
m So I tifdng iJng vdi not Do (C). - t ^0 4 not Re (D).
not Mi (E).
not Fa (F). . . not Sol (G).
• So 6 ttfdng iJng vdi not La (A).
V''
• So 7 tifdng iJng vdi not Si (B).
• So 8 tu-dng u-ng vdi not Do (C).
• So 9 tu-dng u-ng vdi not Re (D).
• Va cu" tiep tuc nhu" vay.
Vi du, mot day gom 6 so Fibonacci dau tien 1, 2, 3, 5, 8 va 13 tu-dng iJng vdi day cdc not nhac C, D, E, G, C va A.
De xay difng nhip dieu vanxd ngu'di ta di tim cac doan nhac c6 tinh chu ky trong ban vanxd Fibonacci. Doan nhac difdc gpi la c6 tinh chu ky neu nhif c6 the chia no ra thanh 2 doan giong het nhau. Vi du, doan nhac GCAGCA la doan c6 tinh chu ky, vi no gom hai doan giong nhau GCA.
Day so* Fibonacci trong thi trifcTng chu'ng ichoan