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III-/ CC DNG TON V DY S V PHNG PHP GII
III-/ CC DNG TON V DY S V PHNG PHP GII
1. Cc kin thc cn nh:
Trong dy s t nhin lin tip c mt s chn li n mt s l ri li n mt s chn V vy, nu:
Dy s bt u t s l v kt thc l s chn th s lng cc s l bng s lng cc s chn.
Dy s bt u t s chn v kt thc cng l s l th s lng cc s chn bng s lng cc s l.
Nu dy s bt u t s l v kt thc cng l s l th s lng cc s l nhiu hn cc s chn l 1 s.
Nu dy s bt u t s chn v kt thc cng l s chn th s lng cc s chn nhiu hn cc s l l 1 s.
a. Trong dy s t nhin lin tip bt u t s 1 th s lng cc s trong dy s chnh bng gi tr ca s cui cng ca s y.
b. Trong dy s t nhin lin tip bt u t s khc s 1 th s lng cc s trong dy s bng hiu gia s cui cng ca dy s vi s lin trc s u tin.
2. Cc loi dy s:
+ Dy s cch u:
- Dy s t nhin.
- Dy s chn, l.
- Dy s chia ht hoc khng chia ht cho mt s t nhin no .
+ Dy s khng cch u.
- Dy Fibonacci hay tribonacci.
- Dy c tng (hiu) gia hai s lin tip l mt dy s.
+ Dy s thp phn, phn s:
3. Cch gii cc dng ton v dy s:
Dng 1: in thm s hng vo sau, gia hoc trc mt dy s
Trc ht ta cn xc nh li quy lut ca dy s:
+ Mi s hng (k t s hng th 2) bng s hng ng trc n cng (hoc tr) vi mt s t nhin a.
+ Mi s hng (k t s hng th 2) bng s hng ng trc n nhn (hoc chia) vi mt s t nhin q khc 0.
+ Mi s hng (k t s hng th 3) bng tng 2 s hng ng lin trc n.
+ Mi s hng (k t s hng th 4) bng tng ca s hng ng trc n cng vi s t nhin d ri cng vi s th t ca s hng y.
+ S hng ng sau bng s hng ng trc nhn vi s th t ca n.
+ Mi s hng (k t s hng th 2) tr i u bng a ln s lin trc n.
+ Mi s hng (k t s hng th 2) tr i, mi s lin sau bng a ln s lin trc n cng (tr ) n (n khc 0)................................Cc v d:
Bi 1: in thm 3 s hng vo dy s sau:
1, 2, 3, 5, 8, 13, 21, 34
Mun gii c bi ton trn trc ht phi xc nh quy lut ca dy s nh sau:
Ta thy: 1 + 2 = 3
3 + 5 = 8
2 + 3 = 5
5 + 8 = 13
Dy s trn c lp theo quy lut sau: K t s hng th 3 tr i mi s hng bng tng ca hai s hng ng lin trc n.Ba s hng tip theo l: 21 + 34 = 55;34 + 55 = 89; 55 + 89 = 144 Vy dy s c vit y l: 1, 2, 3, 5, 8, 13, 34, 55, 89, 144Bi 2: Vit tip 3 s hng vo dy s sau:1, 3, 4, 8, 15, 27
Ta nhn thy:8 = 1 + 3 + 4
27 = 4+ 8 + 15
15 = 3 + 4 + 8
T ta rt ra c quy lut ca dy s l: Mi s hng (k t s hng th 4) bng tng ca ba s hng ng lin trc n.
Vit tip ba s hng, ta c dy s sau: 1, 3, 4, 8, 15, 27, 50, 92, 169.
Bi 3: Tm s hng u tin ca cc dy s saubit rng mi dy s c 10 s hng.
a), , 32, 64, 128, 256, 512, 1024
b)..., ..., 44, 55, 66, 77, 88, 99, 110
Gii:
a). Ta nhn xt:
S hng th 10 l: 1024 = 512 x 2S hng th 9 l: 512 = 256 x 2
S hng th 8 l: 256 = 128 x 2
S hng th 7 l: 128 = 64 x 2
..
T ta suy lun ra quy lut ca dy s ny l: mi s hng ca dy s gp i s hng ng lin trc .
Vy s hng u tin ca dy l: 1 x 2 = 2.
b). Ta nhn xt:
S hng th 10 l: 110 = 11 x 10
S hng th 9 l: 99 = 11 x 9
S hng th 8 l: 88 = 11 x 8
S hng th 7 l: 77 = 11 x 7
..
T ta suy lun ra quy lut ca dy s l: Mi s hng bng s th t ca s hng y nhn vi 11.
Vy s hng u tin ca dy l: 1 x 11 = 11.
Bi 4: Tm cc s cn thiu trong dy s sau:
a. 3, 9, 27, ..., ..., 729.
b. 3, 8, 23, ..., ..., 608.
Gii:
Mun tm c cc s cn thiu trong mi dy s, cn tim c quy lut ca mi dy s .
a. Ta nhn xt:3 x 3 = 9
9 x 3 = 27
Quy lut ca dy s l: K t s hng th 2 tr i, mi s hng gp 3 ln s lin trc n.
Vy cc s cn thiu ca dy s l:
27 x 3 = 81; 81 x 3 = 243; 243 x 3 = 729 (ng).
Vy dy s cn thiu hai s l: 81 v 243.
b. Ta nhn xt:3 x 3 1 = 8 ;
8 x 3 1 = 23.
..........................................
Quy lut ca dy s l: K t s hng th 2 tr i, mi s hng bng 3 ln s lin trc n tr i 1. V vy, cc s cn thiu dy s l:
23 x 3 - 1 = 68;68 x 3 1 = 203;203 x 3 1 = 608 (ng).
Dy s cn thiu hai s l: 68 v 203.Bi 5: Lc 7h sng, mt ngi i t A n B v mt ngi i t B n A; c hai cng i n ch ca mnh lc 2h chiu. V ng i kh dn t A n B; nn ngi i t A, gi u i c 15km, c mi gi sau li gim i 1km. Ngi i t B gi cui cng i c 15km, c mi gi trc li gim 1km. Tnh qung ng AB.
Gii:
2 gi chiu l 14h trong ngy.
2 ngi i n ch ca mnh trong s gi l:
14 7 = 7 gi.
Vn tc ca ngi i t A n B lp thnh dy s:
15, 14, 13, 12, 11, 10, 9.
Vn tc ca ngi i t B n A lp thnh dy s:
9, 10, 11, 12, 13, 14, 15.
Nhn vo 2 dy s ta nhn thy u c cc s hng ging nhau vy qung ng AB l: 9 + 10 + 11 + 12 + 13 + 14 + 15 = 84 p s: 84km.
Bi 6: in cc s thch hp vo trng sao cho tng s 3 lin tip u bng 2010783998
Gii:
Ta nh s th t cc nh sau:
783998
12345678910
Theo iu kin ca bi ta c:
783 + 7 + 8 = 2010.
7 + 8 + 9 = 2010.
Vy 9 = 783; t ta tnh c:
8 = 5 = 2 = 2010 - (783 + 998) = 229
7 = 4 = 1 = 998
3 = 6 = 783.
in cc s vo ta c dy s:
998229783998229783998229783998
Mt s lu khi ging dy Ton dng ny l: Trc ht phi xc nh c quy lut ca dy l dy tin, dy li hay dy s theo chu k. T m hc sinh c th in c cc s vo dy cho.
* Bi tp t luyn:
Bi 1: 13, 19, 25, 31,,
Dy s va c vit ra
Ba s vit tip l ba s no?
S no suy ngh thp cao?
em, bn lm sao k lin?
Bi 2: Tm v vit ra cc s hng cn thiu trong dy s sau:
a. 7, 10, 13,, , 22, 25.
b. 103, 95, 87,, , ...., 55, 47.
Bi 3: in s thch hp vo trng, sao cho tng cc s 3 lin nhau bng:
a. n = 14,5
2,78,5
b. n = 23,4
8,77,6
Bi 4: Cho dy phn s sau:
; ; ;
a) Hy vit tip s hng th nm ca dy theo ng quy lut?
b) Chng t dy trn l mt dy xp theo th t tng dn?
Bi 5: Vit tip ba s hng vo dy s sau :
a) 1; 3; 4; 7; 11; 18;...
b) 0; 2; 4; 6; 12; 22;...
c) 0 ; 3; 7; 12;...
d) 1; 2; 6; 24;...
Dng 2: Xc nh s A c thuc dy cho hay khng?
Cch gii ca dng ton ny:
- Xc nh quy lut ca dy;
- Kim tra s A c tho mn quy lut hay khng?
Cc v d: Bi 1: Cho dy s: 2, 4, 6, 8,
a. Dy s c vit theo quy lut no?
b. S 2009 c phi l s hng ca dy khng? V sao?
Gii:
a. Ta nhn thy:
S hng th 1:2 = 2 x 1
S hng th 2:4 = 2 x 2
S hng th 3:6 = 2 x 3
.........
S hng th n:? = 2 x n
Quy lut ca dy s l: Mi s hng bng 2 nhn vi s th t ca s hng y.
b. Ta nhn thy cc s hng ca dy l s chn, m s 2009 l s l, nn s 2009 khng phi l s hng ca dy.Bi 2: Cho dy s: 2, 5, 8, 11, 14, 17,
- Vit tip 3 s hng vo dy s trn?
- S 2009 c thuc dy s trn khng? Ti sao?
Gii:
- Ta thy:8 5 = 3;11 8 = 3;
Dy s trn c vit theo quy lut sau: K t s th 2 tr i, mi s hng bng s hng ng lin trc n cng vi 3.
Vy 3 s hng tip theo ca dy s l: 17 + 3 = 20 ; 20 + 3 = 23 ; 23 + 3 = 26
Dy s c vit y l: 2, 5, 8, 11, 14, 17, 20, 23, 26.
Ta thy: 2 : 3 = 0 d 2 ; 5 : 3 = 1 d 2 ; 8 : 3 = 2 d 2 ; .....
Vy y l dy s m mi s hng khi chia cho 3 u d 2. M:
2009 : 3 = 669 d 2. Vy s 2009 c thuc dy s trn v cng chia cho 3 th d 2.
Bi 3: Em hy cho bit:
a. Cc s 60, 483 c thuc dy 80, 85, 90, hay khng?
b. S 2002 c thuc dy 2, 5, 8, 11, hay khng?
c. S no trong cc s 798, 1000, 9999 c thuc dy 3, 6, 12, 24, gii thch ti sao?
Gii:
a. C 2 s 60, 483 u khng thuc dy cho v:
- Cc s hng ca dy cho u ln hn 60.
- Cc s hng ca dy cho u chia ht cho 5, m 483 khng chia ht cho 5.
b. S 2002 khng thuc dy cho v mi s hng ca dy khi chia cho 3 u d 2, m 2002 chia 3 th d 1.
c. C 3 s 798, 1000, 9999 u khng thuc dy 3, 6, 12, 24, v:
- Mi s hng ca dy (k t s hng th 2) u gp i s hng lin trc nhn n; cho nn cc s hng (k t s hng th 3) c s hng ng lin trc l s chn, m 798 chia cho 2 = 399 l s l.
- Cc s hng ca dy u chia ht cho 3, m 1000 li khng chia ht cho 3.
- Cc s hng ca dy (k t s hng th 2) u chn, m 9999 l s l.
Bi 4: Cho dy s: 1; 2,2; 3,4; ; 13; 14,2.
Nu vit tip th s 34,6 c thuc dy s trn khng?
Gii:
- Ta nhn xt: 2,2 - 1 = 1,2;3,4 - 2,2 = 1,2;14,2 - 13 = 1,2;
Quy lut ca dy s trn l: T s hng th 2 tr i, mi s hng u hn s hng lin trc n l 1,2 n v:
- Mt khc, cc s hng trong dy s tr i 1 u chia ht cho 1,2.
V d:
(13 - 1) chia ht cho 1,2
(3,4 - 1) chia ht cho 1,2
M: (34,6 - 1) : 1,2 = 28 d 0.
Vy nu vit tip th s 34,6 cng thuc dy s trn.
Bi 5: Cho dy s: 1996, 1993, 1990, 1987,, 55, 52, 49.
Cc s sau y c phi l s hng ca dy khng?
100, 123, 456, 789, 1900, 1436, 2009?
Gii:Nhn xt: y l dy s cch u 3 n v.
Trong dy s ny, s ln nht l 1996 v s b nht l 49. Do , s 2009 khng phi l s hng ca dy s cho v ln hn 1996.
Cc s hng ca dy s cho l s khi chia cho 3 th d 1. Do , s 100 v s 1900 l s hng ca dy s .
Cc s 123, 456, 789 u chia ht cho 3 nn cc s khng phi l s hng ca dy s cho. S 1436 khi chia cho 3 th d 2 nn khng phi l s hng ca dy s cho.* Bi tp l luyn:
Bi 1: Cho dy s: 1, 4, 7, 10,
a. Nu quy lut ca dy.
b. S 31 c phi l s hng ca dy khng?
c. S 2009 c thuc dy ny khng? V sao?
Bi 2: Cho dy s: 1004, 1010, 1016,, 2012.
Hi s 1004 v 1760 c thuc dy s trn hay khng?
Bi 3: Cho dy s: 1, 7, 13, 19,,
a. Nu quy lut ca dy s ri vit tip 3 s hng tip theo.
b. Trong 2 s 1999 v 2009 th s no thuc dy s? V sao?
Bi 4: Cho dy s: 3, 8, 13, 18,
C s t nhin no c ch s tn cng l 6 m thuc dy s trn khng?
Bi 5: Cho dy s: 1, 3, 6, 10, 15,, 45, 55,
a. S 1997 c phi l s hng ca dy s ny hay khng?
b. S 561 c phi l s hng ca dy s ny hay khng? Dng 3: Tm s s hng ca dy
* Cch gii dng ny l:
i vi dng ton ny, ta thng s dng phng php gii ton khong cch (ton trng cy). Ta c cng thc sau :
S cc s hng ca dy = s khong cch+ 1.
c bit, nu quy lut ca dy l : Mi s hng ng sau bng s hng lin trc cng vi s khng i d th:
S cc s hng ca dy = ( S hng ln nht S hng nh nht ) : d + 1.
Cc v d:
Bi 1: Cho dy s 11; 14; 17;.....;65; 68.
Hy xc nh dy s trn c bao nhiu s hng?
Li gii :
Ta c : 14 - 11= 3; 17 - 14 = 3;....
Vy quy lut ca dy s l mi s hng ng lin sau bng s hng mg lin trc n cng vi 3. S cc s hng ca dy s l:
( 68 - 11 ) : 3 + 1 = 20 ( s hng )Bi 2: Cho dy s: 2, 4, 6, 8, 10,, 1992
Hy xc nh dy s trn c bao nhiu s hng?
Gii:
Ta thy:
4 2 = 2;
8 6 = 2
6 4 = 2;
Vy, quy lut ca dy s l: Mi s hng ng sau bng mt s hng ng trc cng vi 2. Ni cc khc: y l dy s chn hoc dy s cch u 2 n v.
Da vo cng thc trn:
(S hng cui s hng u) : khong cch + 1
Ta c: S cc s hng ca dy l:
(1992 - 2) : 2 + 1 = 996 (s hng).
Bi 3: Cho 1, 3, 5, 7, l dy s l lin tip u tin; hi 1981 l s hng th bao nhiu trong dy s ny? Gii thch cch tm?
( thi hc sinh gii bc tiu hc 1980 1981)
Gii:
Ta thy:
S hng th nht bng:1 = 1 + 2 x 0
S hng th hai bng: 3 = 1 + 2 x 1
S hng th ba bng:5 = 1 + 2 x 2
Cn s hng cui cng: 1981 = 1 + 2 x 990
V vy, s 1981 l s hng th 991 trong dy s .
Bi 4: Cho dy s: 3, 18, 48, 93, 153,
a. Tm s hng th 100 ca dy.
b. S 11703 l s hng th bao nhiu ca dy?
Gii:
a. S hng th nht:3 = 3 + 15 x 0
S hng th hai:18 = 3 + 15 x 1
S hng th ba:48 = 3 + 15 x 1 + 15 x 2
S hng th t:93 = 3 + 15 x 1 + 15 X 2 + 15 x 3
S hng th nm:153 = 3 + 15 x 1 + 15 x 2 + 15 x 3 + 15 x 4
S hng th n:3 + 15 x1 + 15 x 2 +15 x 3 + + 15 x (n - 1)
Vy s hng th 100 ca dy l:
3 + 15 x 1 + 15 x 2 + + 15 x (100 - 1)
= 3 + 15 x (1 + 2 + 3 + + 99) (a v mt s nhn vi mt tng.
= 3 + 15 x (1 + 99) x 99 : 2 = 74253
b. Gi s 11703 l s hng th n ca dy:
Theo quy lut phn a ta c:
3 + 15 x 1 + 15 x 2 + 15 x 3 + x (n 1) = 11703
3 + 15 x (1 + 2 + 3 + + ( n 1))
= 11703
3 + 15 x (1 + n 1) x (n 1) : 2 = 11703
15 x n x (n 1) = (11703 3) x 2
= 23400
n x (n 1) = 23400 : 15
= 1560
Nhn xt: S 1560 l tch ca hai s t nhin lin tip 39 v 40 (39 x 40 = 1560)
Vy, n = 40, s 11703 l s hng th 40 ca dy.
Bi 5: Trong cc s c ba ch s, c bao nhiu s chia ht cho 4?
Li gii:
Ta nhn xt : S nh nht c ba ch s chia ht cho 4 l 100 v s ln nht c ba ch s chia ht cho 4 l 996. Nh vy cc s c ba ch s chia ht cho 4 lp thnh mt dy s c s hng nh nht l 100, s hng ln nht l 996 v mi s hng ca dy ( k t s hng th hai ) bng s hng ng lin trc cng vi 4.
Vy s cc s c ba ch s chia ht cho 4 l :
( 996 100 ) : 4 = 225 ( s )
* Bi tp t luyn:
Bi 1: Cho dy s: 3, 8, 13, 23, ,2008
Tm xem dy s c bao nhiu s hng ?
Bi 2: Tm s s hng ca cc dy s sau:
a. 1, 4, 7, 10, ,1999.
b. 1,1; 2,2; 3,3; ...; 108,9; 110,0.Bi 3: Xt dy s: 100, 101, , 789.
Dy ny c bao nhiu s hng?Bi 4: C bao nhiu s khi chia cho 4 th d 1 m nh hn 2010 ?
Bi 5: Ngi ta trng cy hai bn ng ca mt on ng quc l di 21km. Hi phi dng bao nhiu cy trng trn on ng ? Bit rng cy n trng cch cy kia 5m.
Dng 4: Tm s hng th n ca dy sBi ton 1: Cho dy s: 1, 3, 5, 7,............Hi s hng th 100 ca dy s l s no
Gii:
S khong cch t s u n s hng th 100 l:
98 - 1 = 99
Mi khong cch l
3 - 1 = 5 - 3 = 2
S hng th 100 l
1 + 99 ( 2 = 199
Cng thc tng qut:
S hng th n = s u + khong cch ( (S s hng - 1)
Bi ton 2: Tm s hng th 100 ca cc dy s c vit theo quy lut:
a) 3, 8, 15, 24, 35, (1)
b) 3, 24, 63, 120, 195, (2)
c) 1, 3, 6, 10, 15,. (3)Gii: a) Dy (1) c th vit di dng: 1x3, 2x4, 3x5, 4x6, 5x7,
Mi s hng ca dy (1) l tch ca hai tha s, tha s th hai ln hn tha s th nht 2 n v. Cc tha s th nht lm thnh mt dy: 1, 2, 3, 4, 5, ; Dy ny c s hng th 100 l 100.
S hng th 100 ca dy (1) bng: 100x102 = 10200.
b) Dy (2) c th vit di dng: 1x3, 4x6, 7x9, 10x12, 13x15,
Mi s hng ca dy (2) l tch ca hai tha s, tha s th hai ln hn tha s th nht 2 n v. Cc tha s th nht lm thnh mt dy: 1, 4, 7, 10, 13, ; S hng th 100 ca dy 1, 4, 7, 10, 13, l: 1 + (100 1 ) x 3 = 298.
S hng th 100 ca dy (2) bng: 298 x 300 = 89400.
c) Dy (3) c th vit di dng:
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4 ...S hng th 100 ca dy (3) bng:
* Bi tp t luyn:
Bi 1: Cho dy s : 101, 104, 107, 110, ......
Tm s hng th 1998 ca dy s .
Bi 2: Cho dy s : 5, 8, 11, 14, ......a) Tm s hng th 200 ca dy s.
b) Nu c vit tip th cc s: 1000; 2009; 5000 c l s hng ca dy khng? Ti sao.
Bi 3: Mt bn hc sinh vit lin tip cc s t nhin m khi chia cho 3 th d 2 bt u t s 5 thnh dy s. Vit n s hng th 100 th pht hin vit sai. Hi bn vit sai s no?Dng 5: Tm s ch s ca dy khi bit s s hngBi ton 1: Cho dy s: 1, 2, 3,.......150. Hi vit dy s ny ngi ta phi dng bao nhiu ch s
Gii:
Dy s cho c : ( 9 - 1) : 1 + 1 = 9 s c 1 ch s.
C ( 99 - 10 ) : 1 + 1 = 90 s c 2 ch s
C ( 150 - 100) : 1 + 1 = 51 s c 3 ch s.
Vy s ch s cn dng l :
9 ( 1 + 90 ( 2 + 51 ( 3 = 342 ch s
Bi ton 2: Mt quyn sch c 234 trang. Hi nh s trang quyn sch ngi ta phi dng bao nhiu ch s.
Gii:
nh s trang quyn sch ngi ta phi vit lin tip cc s t nhin t 1 n 234 thnh dy s. Dy s ny c
( 9 - 1) : 1 + 1 = 9 s c 1 ch s
C: ( 99 - 10) : 1 + 1 = 90 s c 2 ch s
C: ( 234 - 100) : 1 + 1 = 135 s c 3 ch s
Vy ngi ta phi dng s ch s l:
9 ( 1 + 90 ( 2 + 135 ( 3 = 594 ch s
* Bi tp t luyn:
Bi 1: Mt bn hc sinh vit lin tip cc s t nhin t 101 n 2009 thnh 1 s rt ln. Hi s c bao nhiu ch s
Bi 2: Trng Tiu hc Thnh Cng c 987 hc sinh. Hi ghi s th t hc sinh trng ngi ta phi dng bao nhiu ch s
Bi 3: Cn bao nhiu ch s nh s trang ca mt cun sch c tt c l:
a) 752 trang.
b) 1251 trang.
Dng 6: Tm s s hng khi bit s ch sBi ton 1: nh s trang 1 quyn sch ngi ta dng ht 435 ch s. Hi quyn sch c bao nhiu trang?Gii:
nh s trang quyn sch , ngi ta phi vit lin tip cc s t nhin bt u t 1 thnh dy s. Dy s ny c
9 s c 1 ch s
c 90 s c 2 ch s
vit cc s ny cn s ch s l
9 ( 1 + 90 ( 2 = 189 ch s
S ch s cn li l:
435 - 189 = 246 ch s
S ch s cn li ny dng vit tip cc s c 3 ch s bt u t 100. Ta vit c
246 : 3 = 82 s
S trang quyn sch l
99 + 82 = 181 ( trang)Bi ton 2:
nh s trang mt cun sch ngi ta phi dng tt c 600 ch s. Hi quyn sch c bao nhiu trang? Gii: 99 trang u cn dng 9x1 + 90x2 = 189 ch s.
999 trang u cn dng: 9x1 + 90x2 + 900x3 = 2889 ch s
V: 189 < 600 < 2889 nn trang cui cng phi c 3 ch s. S ch s nh s cc trang c 3 ch s la: 600 - 189 = 411 (ch s)
S trang c 3 ch s l 411: 3 = 137 trang.
Vy quyn sch c tt c l: 99 + 137 = 236 trang.Bi ton 3: ghi th t cc nh trn mt ng ph, ngi ta dng cc s chn 2, 4, 6, 8 . . . ghi cc nh dy phi v cc s l 1, 3, 5, 7 . . . ghi cc nh dy tri ca ng ph . Hi s nh cui cng ca dy chn trn ng ph l bao nhiu, bit rng khi nh th t cc nh ca dy ny, ngi ta dng 367 lt ch s c thy.
Giai:
S nh c s th t ghi bng 1 ch s chn la: (8 - 2) : 2 + 1 = 4 (nh)
S nh c s th t ghi bng 2 ch s chn la: (98 - 10) : 2 + 1 = 45 (nh)
S lt ch s nh s th t cc nh c 1 v 2 ch s la:
4 + 45 2 = 94 (lt)
S lt ch s nh s th t nh c 3 ch s la: 367 - 94 = 273 (lt)
S nh c s th t 3 ch s la: 273 : 3 = 91 (nh)
Tng s nh cua day chn la: 4 + 45 + 91 = 140 (nh)
S nh cui cng ca dy chn l: (140 - 1) 2 + 2 = 280.
Bi ton 4: Cho day s: 1, 3, 5, 7, ..., n. Hay tim s n s ch s cua day gp 3 ln s cac s hang cua day.
Giai:
tim c s n sao cho s cac ch s cua day gp ba ln s cac s hang cua day o, ta gia s trung binh mi s le lin tip cua day u co 3 ch s. Do o:
- T 1 n 9 gm cac s le co mt ch s la:
(9 - 1): 2 + 1 = 5 (s)
Mi s cn phai vit thm 2 ch s nn s ch s cn phai vit thm la:
2 x 5 = 10 (ch s)
Cac s le gm hai ch s la
(99 - 11): 2 + 1 = 45 (s)
Mi s cn phai vit thm 1 ch s nn s ch s cn phai vit thm la:
1 x 45 = 45 (ch s)
Cac s le gm 3 ch s la:
( 999 - 101) : 2 + 1 = 450 (s)
Cac s co 3 ch s am bao s ch s cua day gp ba ln s s hang cua day o.
T 1001 tr i, mi s cn bt i mt ch s. S ch s cn thm phai bng s ch s cn bt va bng:
10 + 45 = 55 (ch s)
Vi mi s phai bt i 1 ch s nn s cac s le co 4 ch s la:
55 : 1 = 55 (s)
Ta co:
(n - 1001) : 2 + 1 = 55
(n - 1001) : 2 = 55 - 1 = 54
(n - 1001) = 54 x 2 = 108
n = 108 + 1001 = 1109
* Bi tp t luyn:
Bi 1: vit dy s t nhin lin tip bt u t 1 ngi ta dng ht 756 ch s. Hi s hng cui cng ca dy s l bao nhiu.
Bi 2: ghi s th t hc sinh ca 1 trng Tiu hc, ngi ta phi dng 1137 ch s. Hi trng c bao nhiu hc sinh?
Bi 3: Tnh s trang ca mt cun sch. Bit rng nh s trang ca cun sch ngi ta phi dng 3897 ch s?
Bi 4: anh s trang cua mt quyn sach, ngi ta phai dung trung binh mi trang 4 ch s. Hoi quyn sach o co bao nhiu trang?Dng 7: Tm ch s th n ca dy
Bi ton 1: Cho dy s 1, 2, 3,..... Hi ch s th 200 l ch s no ?
Gii:
Dy s cho c 9 s c 1 ch s
C 90 s c 2 ch s
vit cc s ny cn
9 ( 1 + 90 ( 2 = 189 ch s
S ch s cn li l
200 - 189 = 11 ch s
S ch s cn li ny dng vit cc s c 3 ch s bt u t 100. Ta vit c
11 : 3 = 3 s (d 2 ch s)Nn c 3 s c 3 ch s c vit lin tip n
99 + 3 = 102
Cn d 2 ch s dng vit tip s 103 nhng ch vit c 10. Vy ch s th 200 ca dy l ch s 0 ca s 103.
Bi ton 2: Cho dy s 2, 4, 6, 8, ..... Hi ch s th 2010 ca dy l ch s no?Gii:
Dy s cho c 4 s c 1 ch s
C (98 - 10) : 2 + 1 = 45 s c 2 ch s C (998 - 100) : 2 + 1 = 450 s c 3 ch s
vit cc s ny cn: 4 ( 1 + 45 ( 2 + 450 x 3 = 1444 ch s
S ch s cn li l:2010 - 1444 = 566 ch s
S ch s cn li ny dng vit cc s c 4 ch s bt u t 1000. Ta vit c:
566 : 4 = 141 s (d 2 ch s)
Nn c 141 s c 4 ch s c vit , s c 4 ch s th 141 l:
(141 - 1) x 2 + 1000 = 1280
Cn d 2 ch s dng vit tip s 1282 nhng mi ch vit c 12. Vy ch s th 2010 ca dy l ch s 2 hng trm ca s 1282.
Bi ton 3: Tim ch s th 2010 phn thp phn cua s thp phn bng phn s .
Gii:
S thp phn bng phn s la: 1 : 7 = 0,14285714285......
y la s thp phn v han tun hoan. Ta thy c 6 ch s thi lp thanh 1 nhom 142857. Vi 2010 ch s thi co s nhom la:
2010 : 6 = 335 (nhom). Vy ch s th 2010 phn thp phn cua s thp phn bng phn s la ch s 7.
Bi ton 4: Cho 1 s c 2 ch s, mt dy s c to nn bng cch nhn i ch s hng n v ca s ny ri cng vi ch s hng chc, ghi li kt qu; tip tc nh vy vi s va nhn c ... (V d c th l dy: 59, 23, 8, 16, 13, ... ). Tm s th 2010 ca dy nu s th nht l 14.
Gii:
Ta lp c dy cc s nh sau:
14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, .....
Ta thy c ht 18 s th dy cc s li c lp li nh dy 18 s u.
Vi 2010 s th c s nhm l:
2010 : 18 = 111 nhm (d12 s)
12 s d l cc s ca nhm th 112 ln lt l: 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1. Vy s th 2010 ca dy l s 1.* Bi tp t luyn:
Bi 1: Cho dy s: 2, 5, 8, 11,.......Hy tm ch s th 200 ca dy s .
Bi 2: Cho dy s: 2, 4, 6, 8, ..... Bn Minh tm c ch s th 2010 ca dy l ch s 0, hi bn tm ng hay sai?
Bi 3: Ban Minh ang vit phn s di dang s thp phn. Thy ban Thng sang chi, Minh lin d: ban tim c ch s th 100 phn thp phn cua s thp phn ma t ang vit. Thng nghi 1 ti ri tra li ngay: o la ch s 6. Em hay cho bit ban Thng tra li ung hay sai?
Dng 8: Tm s hng th n khi bit tng ca dy s
Bi ton 1: Cho dy s: 1, 2, 3, ......., n. Hy tm s n bit tng ca dy s l 136
Gii:
p dng cng thc tnh tng ta c :
1 + 2 + 3 +........+ n =136
Do : (1 + n ) ( n = 136 ( 2
= 17 ( 8 ( 2
= 16 ( 17
Vy n = 16
Bi ton 2: Cho dy s: 21, 22, 23, ......, n
Tm n bit: 21 + 22 + 23 + ..........+ n = 4840
Gii:
Nu cng thm vo tng trn tng ca cc s t nhin lin tip t 1 n 20 ta c tng sau:
1 + 2 + 3 +..........+ 21 + 22 + 23 +.........+ n
p dng cng thc tnh tng ta c
(1 + n) ( n : 2 = 1 + 2 + ....+ 20 + 4840
= ( 1 + 20) ( 20 : 2 + 4840
= 210 + 4840 = 5050
( 1+ n) ( n = 5050 ( 2
= 10100
= 101 ( 100
Vy n = 100
* Bi tp t luyn:
Bi 1: Cho bit: 1 + 2 + 3 +........+ n = 345. Hy tm s n.Bi 2: Tm s n bit rng
98 + 102 +........+ n = 15050
Bi 3: Cho dy s 10, 11, 12, 13, , x. Tm x tng ca dy s trn bng 5106
Dng 9: Tnh tng ca dy s
Cc bi ton c trnh by chuyn ny c phn ra hai dng chnh, l:
Dng th nht: Dy s vi cc s hng l s nguyn, phn s (hoc s thp phn) cch u
Dng th hai: Dy s vi cc s hng khng cch u.
Dng 1: Dy s m cc s hng cch u.Xut pht t mt bi Ton nh sau:
Tnh: A = 1 + 2 + 3 + ... + 98 + 99 + 100
Ta thy tng A c 100 s hng, ta chia thnh 50 nhm, mi nhm c tng l 101 nh sau:
A = (1 + 100) + (2 + 99) + (3 + 98) + ... + (50 + 51) = 101 + 101 + ... + 101 = 50 x 101 = 5050.
y l bi Ton m lc ln 7 tui nh Ton hc Gaux tnh rt nhanh tng cc s T nhin t 1 n 100 trc s ngc nhin ca thy gio v cc bn b cng lp.
Nh vy bi ton trn l c s u tin chng ta tm hiu v khai thc thm rt nhiu cc bi tp tng t, c a ra nhiu dng khc nhau, c p dng nhiu th loi ton khc nhau nhng ch yu l: tnh ton, tm s, so snh, chng minh. gii quyt c cc dng ton chng ta cn phi nm c quy lut ca dy s, tm c s hng tng qut, ngoi ra cn phi kt hp nhng cng c gii ton khc nhau na.
Cch gii:
Nu s hng ca dy s cch u nhau th tng ca hai s hng cch u u v s hng cui trong dy s bng nhau. V vy:
Tng cc s hng ca dy bng tng ca mt cp hai s hng cch u s hng u v cui nhn vi s hng ca dy chia cho 2.
Vit thnh s :
Tng ca dy s cch u = (s u + s cui) x (s s hng : 2)
T s trn ta suy ra:
S u ca dy = tng x 2 : s s hng s hng cui.
S cui ca dy = tng x 2 : s s hng s u.
Sau y l mt s bi tp c phn thnh cc th loi, trong phn thnh hai dng trn:
Bi 1: Tnh tng ca 19 s l lin tip u tin.
Gii:
19 s l lin tip u tin l:
1, 3, 5, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37.
Ta thy:1 + 37 = 38
;5 + 33 = 38
1 + 35 = 38
;7 + 31 = 38
Nu ta sp xp cc cp s t hai u s vo, ta c cc cp s u c tng s l 38.
S cp s l:
19 : 2 = 9 (cp s) d mt s hng.
S hng d ny l s hng chnh gia dy s v l s 19. Vy tng ca 19 s l lin tip u tin l:
39 x 9 + 19 = 361
p s: 361.
Nhn xt: Khi s s hng ca dy s l (19) th khi sp cp s s d li s hng chnh ga v s l khng chia ht cho 2, nn dy s c nhiu s hng th vic tm s hng cn li s rt kh khn. Vy ta c th lm cch 2 nh sau:
Ta b li s hng u tin l s 1 th dy s c: 19 - 1 = 18 (s hng)
Ta thy:3 + 37 = 40;7 + 33 = 40
5 + 35 = 40;9 + 31 = 40
Khi , nu ta sp xp cc cp s t 2 u dy s gm 18 s hng vo th c cc cp s c tng l 40.
S cp s l:
18 : 2 = 9 (cp s)
Tng ca 19 s l lin tip u tin l:
1 + 40 x 9 = 361
Ch : Khi s hng l s l, ta li mt s hng 2 u dy s (s u, hoc s cui) cn li mt s chn s hng ri sp cp; ly tng ca mi cp nhn vi s cp ri cng vi s hng li th c tng ca dy s.Bi 2: Tnh tng ca s t nhin t 1 n n.
Gii:
Ghp cc s: 1, 2, , n 1, n thnh tng cp (khng sp th t) : 1 vi n, 2 vi (n 1), 3 vi (n 2),
Khi n chn, ta c S = n x (n + 1) : 2
Khi n l, th n 1 chn v ta c:
1 + 2 + + (n 1) = (n 1) x n : 2
T ta cng c:
S = (n 1) x n : 2 + n
= (n - 1) x n : 2 + 2 x n : 2
= [(n 1) x n + 2 x n] : 2
= (n 1 + 2) x n : 2
= n x (n + 1) : 2Khi hc sinh lm quen v thc hin thnh tho th hng dn hc sinh p dng cng thc lun m khng cn nhm thnh cc cp s c tng bng nhau.
Tng ca dy s cch u = (s u + s cui) x s s hng : 2Bi 3: Tnh E = 10,11 + 11,12 + 12,13 + ...+ 98,99 + 100Li gii
Ta c th a cc s hng ca tng trn v dng s t nhin bng cch nhn c hai v vi 100, khi ta c:
100 x E = 1011 + 1112 + 1213 + ... + 9899 + 1000 p dng cng thc tnh tng ta tnh c tng l E = 4954,95Hoc gii nh sau:
Ta thy: 11,12 - 10,11 = 12,13 - 11,12 = ... = 1,01
Vy y l dy s cch u 1,01 n v.
Dy s c s s hng l: (100 - 10,11): 1,01 + 1 = 90 s hng
Tng ca dy s l: (10,11 + 100) x 90: 2 = 4954,95
Bi 4: Cho dy s: 1, 2, 3, 195. Tnh tng cc ch s trong dy?
Gii:
Ta vit li dy s v b sung thm cc s: 0, 196, 197, 198, 199 vo dy:0, 1, 2, 3, , 9
10, 11, 12, 13, , 19
.....................
90, 91, 92, 93, , 99
100, 101, 102, 103, , 109 .............
V c 200 s v mi dng c 10 s, nn c 200 : 10 = 20 (dng)
Tng cc ch s hng n v trong mi dng l:
1 + 2 + 3 + + 9 = 9 x 10 : 2 = 45
Vy tng cc ch s hng n v l:
45 x 20 = 900
Tng cc ch s hng chc trong 10 dng u u bng tng cc ch s hng chc trong 10 dng sau v bng:
1 x 10 + 2 x 10 + + 9 x 10 = (1 + 2 + + 9) x 10 = 45 x 10 = 450
Vy tng cc ch s hng chc l:
450 x 2 = 900
Ngoi ra d thy tng cc ch s hng trm l: 10 x 10 = 100.
Vy tng cc ch s ca dy s ny l:
900 + 900 + 100 = 1900
T suy ra tng cc ch s ca dy ban u l:
1900 (1 + 9 + 6 + 1 + 9 + 7 + 1 + 9 + 8 + 1 + 9 + 9) = 1830
Trong Ton hc ni ring v trong khoa hc ni chung, chng ta thng nh vo suy lun quy np khng hon ton m pht hin ra nhng kt lun (gi l gi thuyt) no . Sau chng ta s dng suy lun din dch hoc quy np hon ton kim tra s ng n ca kt lun . Khi dy hc tiu hc, iu ni trn cng c lu .
Bi 5: Tnh tng tt c s thp phn c phn nguyn l 9, phn thp phn c 3 ch s:
Gii:
Cc s thp phn c phn nguyn l 9, phn thp phn c 3 ch s l:
9,000; 9,001; 9,002; 9,003; 9,004; 9,005; 9,006; 9,007; 9,008; ; 9,999 tc l c 1000 s.
Tng tt c cc s ca dy s trn l:
(9,000 + 9,999) x 1000: 2 = 9499,5
p s: 9499,5
Bi 6: Phi thm vo tng cc s hng trong dy s: 2, 4, 6, 8, ..., 246 t nht bao nhiu n v c s chia ht cho 100?
Gii:
y l dy s chn lin tip hay dy s cch u 2 n v.
Dy s c s s hng l: (246 - 2) : 2 + 1 = 123 s hng.
Tng ca dy s l: (246 + 2) x 123 : 2 = 12252
V 100 - 52 = 48 nn phi thm vo tng ca dy s t nht 48 n v. Dng 2: Dy s m cc s hng khng cch u.Bi ton 1: Tng nhiu phn s c t s bng nhau v mu s ca phn s lin sau gp mu s ca phn s lin trc 2 ln.
V d: .
Cch gii:
Cch 1:
Bc 1: t A =
Bc 2: Ta thy:
Bc 3: Vy A =
A =
A = 1 -
A =
p s: .
Cch 2:
Bc 1: t A =
Bc 2: Ta thy:
.
Bc 3: Vy A =
= 1 - =
Bi ton 2: Tnh tng ca nhiu phn s c t s bng nhau v mu s ca phn s lin sau gp mu s ca phn s lin trc n ln (n > 1).
V d: B =
Cch gii:
Bc 1: Tnh B x n (n = 3)
B x 3 = 3 x
=
Bc 2: Tnh B x n - B
B x 3 - B = -
B x (3 - 1) = -
B x 2 =
B x 2 =
B x 2
B =
B
B
Bi ton 3: Tnh tng ca nhiu phn s c t s l n (n > 0); mu s l tch ca 2 tha s c hiu bng n v tha s th 2 ca mu phn s lin trc l tha s th nht ca mu phn s lin sau:
V d 1: A =
Cch gii:
A =
=
=
=
V d 2:
B =
Cch gii:
B =
B =
=
=
* Bi tp t luyn:
Bi 1: Tnh tng:
a) Ca tt c cc s l b hn 100
b) 1 + 4 + 9 + 16 + + 169
Bi 2:
a) Tnh nhanh tng ca tt c cc s c 3 ch s.
b) 1, 2, 3, 6, 12, 24, 48, 96, 192, 384.
Dy s trn c mi s hng
Tng bao nhiu, mi bn tnh nhanh
em, ch, anh
Tm ra phng php tnh nhanh mi ti.
Bi 3: Tnh nhanh:
a)
b)
c)
Bi 4: + + + + + + = ?
Php cng phn s kh g?
K s hng ra th ung cng
Cch g ai t ai thng
Cng nhanh p ng li khng tn gi
bn hin em th
ai ai bit y nh gii mau.
Bi 5: Hy tnh tng ca cc dy s sau:
a) 1, 5, 9, 13, 17, Bit dy s c 80 s hng.
b) ..., 17, 27, 44, 71, 115. Bit dy s c 8 s hng.
Bi 6: Tnh nhanh:
a) 1,27 + 2,77 + 4,27 + 5,77 + 7,27 + + 13,27 + 14,77
b) 0,1 + 0,2 + 0,3 + 0,4 + + 0,9 + 0,10 + 0,11 + 0,12 + + 0,19.Bi 7: Cho dy s:
a) Hy tnh tng ca 10 s hng u tin ca dy s trn.
b) S c phi l mt s hng ca dy s trn khng? V sao?
Dng 10: Dy ch
Khc vi cc dng ton khc, ton v dng dy ch khng i hi hc sinh phi tnh ton phc tp. Ngc li gii nhng bi ton dng ny, i hi hc sinh phi bit vn dng sng to nhng kin thc ton hc n gin, nhng hiu bit v x hi, t m vn dng dng ton ny vo trong i sng hng ngy v cc mn hc khc.
Cc v d:Bi ton 1: Ngi ta vit lin tip nhm ch: HOCSINHGIOITINH thnh mt dy ch lin tip: HOCSINHGIOITINHHOCSINHGIOI hi ch ci th 2009 ca dy l ch ci no?
Gii:
Ta thy mi nhm ch: HOCSINHGIOITINH gm 15 ch ci. Gi s dy ch c 2009 ch ci th c:
2009 : 15 = 133 (nhm) v cn d 14 ch ci.
Vy ch ci th 2009 ca dy ch HOCSINHGIOITINH l ch N ca ting TINH ng v tr th 14 ca nhm ch th 134.
Bi ton 2: Mt ngi vit lin tip nhm ch THIXAHAIDNG thnh dy THIXAHAIDNGTHIXAHAIDNG Hi:
a. Ch ci th 2002 trong dy ny l ch g?
b. Nu ngi ta m c trong dy s c 50 ch H th dy c bao nhiu ch A? Bao nhiu ch N?
c. Bn Hi m c trong dy c 2001 ch A. Hi bn y m ng hay m sai? Gii thch ti sao?
d. Ngi ta t mu cc ch ci trong dy theo th t: XANH, , TM, VNG, XANH, , TM, hi ch ci th 2001 trong dy c t mu g?
Gii:
a. Nhm ch THIXAHAIDNG c 13 ch ci:
2002 : 13 = 154 (nhm)
Nh vy, k t ch ci u tin n ch ci th 2002 trong dy, ngi ta vit 154 ln nhm THIXAHAIDNG, vy ch ci th 2002 trong dy l ch G ca ting DNG.
b. Mi nhm ch THIXA HAIDNG c 2 ch H v cng c 2 ch A v 1 ch N. V vy, nu ngi ta m c trong dy c 50 ch H th tc l ngi vit 25 ln nhm nn dy phi c 50 ch A v 25 ch N.
c. Bn m sai, v s ch A trong dy phi l s chn.
d. Ta nhn xt:
+ 2001 chia cho 4 th d 1.
+ Nhng ch ci trong dy c s th t l chia cho 4 th d 1 th c t mu XANH.
Vy ch ci th 2001 trong dy c t mu XANH.
Bi ton 3: Bn Hi cho cc vin bi vo hp ln lt theo th t l: bi xanh, bi , bi vng ri li n bi xanh, bi , bi vng ... c nh vy. Hi:
a) Vin bi th 100 c mu g?
b) Mun c 10 vin bi th phi b vo hp t nht bao nhiu vin bi?
Gii:a) Ta thy, c 3 vin bi th lp thnh 1 nhm mu: xanh, , vng. 100 vin bi th c s nhm l: 100 : 3 = 33 nhm (d 1 vin bi)
Nh vy, bn Hi cho vo hp c 33 nhm, cn d 1 vin ca nhm th 34 v l vin bi u tin ca nhm ny. Vy vin bi th 100 c mu xanh.
b) Mt nhm th c 3 vin bi, mun c 10 vin bi th cn b vo hp:
3 x 10 = 30 vin bi. Nhng vin bi mu l vin bi th 2 ca nhm. Vy cn b vo hp t nht s vin bi l: 30 - 1= 29 vin.* Bi tp t luyn:
Bi 1: Mt ngi vit lin tip nhm ch: TOANNAM thnh dy: TOANNAMTOANNAMTOAN Hi:
a. Ch ci th 2010 trong dy l ch g?
b. Nu ngi ta m c trong dy c 50 ch N th dy c bao nhiu ch A? Bao nhiu ch O?
c. Mt ngi m c trong dy c 2009 ch A, hi ngi m ng hay sai? Gii thch ti sao?
d. Ngi ta t mu cc ch ci trong dy theo th t XANH, , TM, VNG, XANH, , TM hi ch ci th 2009 trong dy c t mu g?
Bi 2: Ngi ta vit cc ch ci D, A, Y, T, O, T, H, O, C, T, O, T, thnh dy: DAYTOTHOCTOTDAYTOT bng 3 mu xanh, , tm, mi ting mt mu.Hi ch ci th 2010 l ch ci g? Mu g?Bi 3: Bn Dng vit lin tip cc nhm ch DIENBIENPHU thnh dy: DIENBIENPHUDIENBIENPHU ... Hi:
a) Ch ci th 1954 l ch g?
b) Nu trong dy vit c 2010 ch E th c bao nhiu ch H?
Bi 4: Mt ngi vit lin tip nhm ch TOQUOCVIETNAM thnh dy TOQUOCVIETNAM TOQUOCVIETNAM Hi:
a) Ch ci th 1975 trong dy l ch g?
b) Ngi ta m c trong dy c 50 ch T th dy c bao nhiu ch O? Bao nhiu ch I?
c) Bn An m c trong dy c 1945 ch O. Hi bn y m ng hay sai? V sao?
d) Ngi ta t mu vo cc ch ci trong dy trn theo th t: xanh, , tm, vng, xanh, , tm, vng, Hi ch ci th 2010 c t mu g?
4- Mt s lu khi gii ton v dy s
Trong bi ton v dy s thng ngi ta khng cho bit c dy s (v dy s c nhiu s khng th vit ra ht c) v vy, phi tm ra c quy lut ca dy (m c rt nhiu quy lut khc nhau) mi tm c cc s m dy s khng cho bit. l nhng quy lut ca dy s cch u, dy s khng cch u hoc da vo du hiu chia ht tm ra quy lut.
dng 2: Mun kim tra s A c tho mn quy lut ca dy cho hay khng? Ta cn xem dy s cho trc v s cn xc nh c cng tnh cht hay khng? (C cng chia ht cho mt s no hoc c cng s d) th s thuc dy cho.
dng 3 v 4: Hc sinh phi c t tm ra cng thc tng qut, vn dng mt cch thnh tho v bit bin i cng thc lm cc bi ton khc. dng 9: C cc yu cu:
+ Tm tng cc s hng ca dy.
+ Tnh nhanh tng.
Khi gii: Sau khi tm ra quy lut ca dy, ta sp xp cc s theo tng cp sao cho c tng u bng nhau, sau tm s cp ri tm tng cc s hng ca dy. Ch : Khi tm s cp s m cn d mt s hng th khi tm tng ta phi cng s d vo.
Nu tnh nhanh tng ca cc phn s phi da vo tnh cht ca phn s.
dng 10: l dy ch khi gii phi da vo quy lut ca dy, sau c th xem mi nhm ch c tt c bao nhiu ch ri i tm c tt c bao nhiu nhm v chnh l phn tr li ca bi ton.
PAGE 20
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