1.8 ASU equ (Progressions of numbers :)

 

 

PɼV DQzAiP ĸU qĪ.

 

ĸ 1 : ê ûvAz g 10,000 qɢgAz wAiĪ v VAi w PqwgAz Mj. EzP Pɮ DAiU:

 

1.      ê PAz gƥAiAv Pq Aiĸwj. EzP ûv MvۣAi ? RArv E. KPAzg At wj Īig 28 U P.(10,000/365).

2.      ê wâ z PĸASAiĵ gƥU PqwgAz sĪ.(1 1 g. 2 2 g, 3 3 g. . . . . . U) DU wj J P ?

3.      ê z 1 g. Pl A wâU A Pl tz Jgqg Pqīgzg,( 1 1 g. 2 2 g, 3 4 g 4 8g . . . . U) wj J P?

 

PƣAi Jgq AzsU wj Pz U AS PAq rAiĪz U ?

 

ĸ 2 : ê MAz 70 Q.. zgz P gø sU AiĸwgAz sĪ. z UAmAiİ 16 Q. zg Zwj. A w UAmAiĮ U MAzAz Q..Av PrAiiUĪzzg. zsAi CAw Av vĥ U J Ai P ?

F jwAi, vfêz ĸUU Utv U Ai iqvzAz wAiĪ.

 

18.1 rU (Sequence) :

1.8.1 Gz 1 : Aiİ J vgUwU gAiĮ ýzg U gAiwj ?

3, 10, 4, 1, 12, 8, 7, 5, 6, 2, 9, 11 - JAz gAiwg ?

E zV 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12 - JAz gAiwj.

 

1.8.1 Gz 2 : 2006 E dj wAUgĪ DvgU vjPU gAiĨPzg, U gAiwj : 1, 8, 15, 22, 29 JAz gAiwj

 

ð Jgq AzsU K irj ? UjzAi êAz PP zgV, ASU gɢj.

z Azsz 1 jAz DgA, MAzAz CAPAiģ A ASU Pr, A AS gɢj. 12 Dzq 蹢j. KP? Cz Aiİ EgĪ PƣAi vgUw.

Jgq Azsz 2006 E dj wAU z gg 1 vjP. DzjAz z 1 gz Aa ASU 7 Pr Jߪ AiĪP̣ĸgV A ASU gɢj. PƣAi AS 31QAv Pr EgĪAv rPArj. KPAzg dj wAU 31 ivgvz.

 

1.8.1 Gz 3 : F nAiģ UĤ 2, 4, 6, 8, 10, 12 . . . . . .

Ez Aii n? Ez ĸASU n v Ez VAiĪz E.

 

S : MAz rAi (sequence) AiĪP̣ĸgV UƽgĪ ASU UtVgvz. rAiİ w CA rAi z (Term) Vgvz.

rAi zU U avê, T1 T2T3T4T5 .

 

zU P AS ==

1

2

3

4

----

n

---

CP ZP aU ===

T1

T2

T3

T4

----

Tn

---

 

MAz rAiģ iV {Tn } JAz avê.

 

Jt¸zz Cx zU AgĪ rAi jv r(Finite sequence).

Jt¸Uz Cx C zU AgĪ r Cjv r(Infinite sequence).

ð z GzguAiİ 12 zU v 2 GzguAiİ 5 zU. Egq jv rU GzguUV.

3 GzguAiiz ASU nAiİ zU. DzjAz Cz MAz Cjv r.

 

1.8.1 Gz 4 : MAz rAi gUAz Prgz.

2/1, 3/2, 4/3, 5/4 . . . .

 

E rAi iz Tn ɯ J ?

 

T1 = (1+1)/1

T2 =(2+1)/2

T3 =(3+1)/3

T4 =(4+1)/4

 

Tn=(n+1)/n, F rAi i zAzV, zv rAi Aiiz z PAqĻrAiħz.

rAi 6 z T6 =(6+1)/6 =7/6

 

1.8.1 ĸ 1 : Tn =2n2+1, Tn=73 Dzg n ɯ PAqĻrj ?

Tn =2n2+1 =73

2n2 =73-1=72

2n2 =72

n2 =36

n = =

zv ASAi sP ASAiizjAz n, zs uAPVgP n=6.

v:

T6 = 2*62+1 = 2*36+1=73

 

1.8.2 tU(Series):

 

S : MAz rAi zU v۪ t (series) Jvê. Ez S Cx Sn Az avê. tAi MAz jv rAi zU v۪Vgvz.

 

Sn = T1 + T2+T3.........Tn

 

Sn- Sn-1=( T1 + T2+T3.........Tn-1+ Tn) -( T1 + T2+T3.........Tn-1)= Tn

 

Sn- Sn-1 = Tn

 

1.8.2 ĸ 1 : Tn ={(-1)n} Dzg, S1 = S3 : S2 = S4 JAz .

 

Tn =(-1)n

 

T1= (-1)1 = -1, T2 = (-1)2 =1, T3 = (-1)3 = -1, T4= (-1)4 = 1

S1 = T1 = -1

S3 = T1 + T2+T3= -1+1-1 = -1

S1 = S3

S2 = T1 + T2 =-1+1 =0

S4 =T1 + T2+T3 +T4= -1+1-1+1 =0

S2 = S4

 

 

1.8.3 iAvg rU(Arithmetic Progression):

 

ð Gzgu 1.8.1.1 g, Aiiz Jgq CP zU v 1 DVz.

Gz.1.8.1.2 g, Jgq CP zU v 7 DVz.

 

S: MAz rAiİ Aiiz Jgq CP zU qī v gAPVzg, D rAi iAvg rAiiVgvzɔ(Arithmetic Progression)(AP). gAP i vVgvz v Cz dAz aUvz.

iAvg rAiİ, Tn+1 Tn =d : Tn-1+d = Tn

iAvg rAiİ z z gAPVz Cz aAz avê.

T1 = a

T2= a+d

T3= T2+d =(a+d)+d = a+2d = a + (3-1)d

T4= T3+d =(a+2d)+d =a+3d= a+(4-1)d

Tn = Tn-1+d = a+(n-1)d. d= (Tn -a)/(n-1)

 

iAvg rAi i z: Tn = a+(n-1)d

 

iAvg rAi igƥ: {a, a+d, a+2d,a+3d , a+(n-1)d}

 

 

1.8.3 ĸ 1 : Sn = 5n2+3n Dzg iAvg rAiģ gɬj.

 

jg:

Sn-1 = 5(n-1)2+3(n-1) = 5(n2 -2n+1) +3n-3

= 5n2-10n+5+3n-3

= 5n2-7n+2

Tn= Sn- Sn-1

= (5n2+3n) (5n2-7n+2)

= 10n-2

T1 = 8

T2 =18

T3 =28

iAvg r: {8,18,28..}

 

v:

S3 = T1 + T2 + T3 =8+18+28 = 54

Sn = 5n2+3n

= 5*32+3*3

= 54

 

 

1.8.3 ĸ 2 : MAz iAvg rAiİ T10 =20 T20 =10 Dzg T30 PAqĻr.

 

jg:

z a v dU PAqĻrAiĨP.

Tn = a+(n-1)d

T10 = a+(10-1)d = a+9d

Dzg, T10= 20 zv

a+9d=20: a=20-9d ====(1)

T20 = a+(20-1)d = a+19d ====(2)

Dzg, T20= 10

(1) v (2)jAz, T20 = a+19d

=20-9d+19d =10

=20+10d =10

10d =(10-20)= -10

d = -1

(1)jAz, a =20-9d

= 20+9 =29

T30 = a+(30-1)d

= 29+29*(-1) = 29-29 =0

 

v:

T10 =29+9*(-1)=20

T20 =29+19*(-1)=10

 

1.8.3 ĸ 3: 5 v 10 zU Cĥv 1:2 DVz, T12 =36 DVgĪ iAvg rAiģ gɬj.

 

jg:

T5 : T10 = 1:2 (i.e T5 /T10 =1/2) zv

2T5 = T10

 

2(a+4d) = (a+9d)

2a+8d = a+9d

a=d.

T12 =36 zv

a+ 11d = 36

a=d DzjAz, 12d =36

d=3

a=d DzjAz, a=3

iAvg r: = 3,6,9,12

 

v:

T5 = 15, T10 =30, 1:2 zv Cĥv.

 

1.8.3 ĸ 4: v15 v Ut 105 DVgĪ iAvg rAi g zU PAqĻrj.

 

jg:

zsz z a DVg.

z z: a-d

3 z: a+d.

g zU v= (a-d)+a+(a+d ) = 3a = 15

a = 5.

g zU Ut = (a-d)*a*(a+d) = a*(a2-d2) =105

a*(a2-d2) =105

5(52-d2) = 105

(25-d2) = 21

-d2 = 21-25

-d2= -4

d2= 4

d =

iAvg rAi zU: 3,5,7 Cx 7,5,3

 

v: 3,5,7 EU v 15, Ut: 105.

 

1.8.4 iAvg tAi zU v(Summation of arithmetic series):

 

FU F CzsAiģz DgAsz rz ĸAi U ZaĪ.

1.8.4 ĸ 1 : ê ûvAz g. 10,000 qɢgAz sĪ v Aw t AzP PqĪ vPP ê Mj. DU U Jgq DAiU.

1. ê PAz gƥAiAv Pq Aiĸwj. EzP ûv MvۣAi ? RArv E. KPAzg At wj Īig 28 U P. ( =28).

2.      ê wâ z PĸASAiĵ gƥU PqwgAz sĪ.(1 1 g. 2 2 g, 3 3 g. . . . . . U) Pqv U ê Mwg ? RArv q KP qĪ:

 

2 DAi AiAv ê 10 U Pl t JUvz?

10 U Pl Ml t = 1+2+3+4+5+6+7+8+9+10 = 55g.

Uzg 100 U Pl t J? Ez ASU vۢAz PAqĻrAiĨP.

DzjAz, FU z n sP ASU v۪ PAqĻrAiĪ v K JAz wAiĪ.

{T} = {1,2,3n}

Sn = 1 + 2 + 3 .+(n-2)+ (n-1) +n(n zU)

+ Sn = n +(n-1)+(n-2) + 3 + 2 +1(wgV gɢz)

==================================

2Sn= (n+1)+(n+1)+(n+1) .. .+(n+1)+(n+1)+(n+1) (n zU)

= n(n+1)

Sn=

F vߥAiV, 10 U Pl t P PĪ:

S10 =10*11/2= 55 g.

FU 100 U PqĪ Ml t: S100 = 100*101/2 = 5050g.

200 U PqĪ Ml t: S200 = 200*201/2 =20,100g.

10,000 g.U wj Pz U PAqĻrAiĪ zs Avg qĪ.

FU zP: S141 = =10,011

DzjAz wj 141 U P.

z n zU v Sn Az avê.

=

S: MAz tAi zU iAvg rAiİzg, D tAiģ iAvg t”(arithmetic series) Jߪg.

Gz: {2,5,8}, {1,4,7,}, {3,7,11}

 

iAvg rAi n zU v PAqĻrAiĪz:

 

{AP}= {a, a+d, a+2d, a+3d .,a+(n-1)d}

Sn= [a+(a+d)+(a+2d)+(a+3d) ..a+(n-1)d]

= [a+a+a .(n ) +d(1+2+3+ . (n-1)]

= na+d[]

na+ (= vz n z (n-1) GAiV)

Sn = na+ = = n*()

= n*()=n*()

 

1.8.4 ĸ2 : 25 zUgĪ MAz iAvg tAiİ zsz z 20 Dzg D tAi zU v PAqĻrj.

 

jg:

 

zv: n=25, T13 =20, S25 PAqĻrAiĨP.

T13 = a+12d

S25 = n*(a+ T25)/2= 25*(a+a+24d)/2

= 25*2*(a+12d)/2

= 25*(a+12d) = 25*20(T13 = a+12d)

= 500

 

1.8.4 ĸ 3 : 4 jAz sUUĪ 101 jAz 201 g gV J sP ASU v PAqĻr.

 

jg:

{AP} = (104,108,112 200}

Sn = 104+108+112+

= 104+(104+4) + (104+8) (104+96) (104, 25 j ģgv DUvz.)(UĤ: 1 z =104, PƣAi z 200 v v = 4)

= 104*25 +4(1+2+3..24)

= (104*25) +4*( )

=2600+1200=3800

 

1.8.4 ĸ 4 : ħAi KP UgĪ t ɼUƼP ê VgAz s. ê z ĵz 23 n֮U vwj. Avg w ĵz ê A ĵz wzQAv 2 n֮U Pr vwj JAzzg, 7 ĵ U wz Ml n֮U AS PAqĻrj.

 

jg:

 

ê w ĵz A ĵz wzQAv 2 n֮ Pr v۪zjAz Cz rAiİz. ê 7 ĵ P vUzPAqzjAz, iAvg tAiİ S7PAq rAiĨP.

{AP} = {23,21,19.) a=23, d = -2

Sn = n*( )

S7 = 7* ( )

= 7*[46-12]/2

= 7*17 = 119

 

ê ir: ê wAiģ vĥ 1000 n֮ vۨPVzg, CzP PUĪ P PAqĻrj.

 

1.8.4 ĸ5: ê MAz 70Q.. zgz P gø sUP. z UAmAiİ UAmU 16 Q.. Uz Zwj. Az w UAmAiİAi MAzAz Q.. U PrAiizg, zs V Pz P PAqĻr.

jg:

P U: (16,15,14, ) MAz iAvg r,

Sn =70 DUĪAv n PAqĻrAiĨP.

E a =16, d = -1

Sn = n*( )

= n*( )

= n*()

= n*()

n*() = 70(Ml zg: 70Q..)

 

(33n-n2 ) = 2*70=140

-n2 +33n -140 =0

n2 -33n +140 =0

(n-5)*(n-28) = 0

n=5 Cx n=28

 

Utvz Pg 2 GvgU (5 v 28) qɢzê. Dzg DgAsz U UAmU 16Q.. DVz, UAmU MAz Q.. U PrAiiUĪU Pz P 5 UAmVAv ZU zs嫮(n=28 Dzg U It(T28= -11) DUvz).

Pz P = 5 UAmU.

 

1.8.4 ĸ6: M gd zAi 2 Aid zg V,v櫣 DU rAiĮ 7 U 80 AidU zg, w J zg aѹgP, ĢުAv ü? (ïw P 126)

 

jg:

 

gd PĹz zg MAz iAvg r,

Sn =70 DUĪAv d PAqĻrAiĨP.

E a =2, n = 7

Sn = n*( )

= 7*( )

= 7*()

= 7*(2+3d) = 80

2+3d = 80/7

3d = (80/7)-2 = (66/7)

gd w 22/7 Aid zg aѸP.

 

1.8.4 ĸ7: M zAi 3 zs z ir, w 2 U aѸv zg 360 U J U z iqvۣ. ïwAi U ü. (ïw P 124)

jg:

 

z irz zs MAz iAvg r,

Sn =360 DUĪAv n PAqĻrAiĨP.

E a =3, d = 2

Sn = n*( )

= n*( )

= n*(3n+2n-2) = n(n+2)

n2+2n =360

n2+2n -360 =0

(n+20)*(n-18) =0

 

n=-20 zs嫮. 360 z iq 18 U P.

 

1.8.5 Uuvg r(Geometric Progression) (GP):

 

PɼV GzguU UĤ:

1. {T}= {2,4,8,16 .}. F rAiİ w z A zz Jgqgz.

Aiiz z = 2* A z Cx = 1/2* A z

zU qī Cĥv= 1:2.

2.      {T}= {27,9,3,1 .}. F rAiİ w z A zz 1/3gz.Aiiz z = 1/3* A z Cx A z = 3* A z. zU qī Cĥv =3:1

S: rAi Aiiz MAz z v Czg A zU Cĥv MAz gAPVzg, Cz Uuvg r(Geometric Progression)(GP) Jvg. F gAP i Cĥv Jvg v Cz r Az avê.

MAz GPAiİ Tn /Tn-1 = gAP

1 Gz. z T3 /T2==2 2 Gz. z T3 /T2= = 1/3

MAz Uuvg rAiİ z z T1 = a i Cĥv r Dzg,

T2= T1*r= ar(2-1)

T3= T2*r= ar*r =ar2= ar(3-1)

T4= T3*r= ar2*r = ar3= ar(4-1)

iV, Tn= ar(n-1) ; Tn= ar(n-1 = ar(n-2)*r=Tn-1*r

Uuvg r igƥ:- {a, ar, ar2, ar3 .. ar(n-1)} of GP.

 

1.8.5 ĸ 1 : MAz Uuvg rAiİ 7 z 4 zz JAlgz v 5 z 12 Dzg, D rAiģ gɬj.

 

jg:

 

Tn = arn-1

T7=a r6 , T4=a r3 Dzg T7= 8T4 zv

a r6= 8 ar3

r3= 8

r=2

T5=a r4

= a 24=16a =12 (zv)

a = =

zv Uuvg r = {, *2, *22 , *23.} = {3/4, 3/2,3,6}

 

Uuvg rAi n zU v PAq rAiĪz:

 

= {a, ar, ar2, ar3 .. ar(n-1)}(n terms)

(1) Sn= a +ar+ar2+ ar3 .. +ar(n-1)

ð Pgt r Az Ut¹zU,

(2) rSn= ar+ar2+ ar3 +ar(n-1)+ arn

 

.(2) jAz (1) PzU,

Sn- rSn=a- arn

Sn(1-r) =a(1- rn)

Sn= a (1- rn) / (1-r) ----- r <1 DzU,

= -a (1- rn) /-(1-r) (CA v bzUgq -1jAz Ut¹)

= a ( rn-1) / (r-1) ----- r >1 DzU,

 

r U Aiizɯ ɯ Egz? ( r=1, r>1,r<1)

1) r=1 DzU, GP = {a ,a,a.a,a.}

2) r<1 DzU,

GzguU, r = = 0.9 DzU, n Cw zq AS DzU, G KUvz?

r2=

0.81

r4=

0.66

r8=

0.43

r16=

0.19

r64=

0.0012

 

F jwAiiV, n ɯ Cw ZzU, rn ƣAiģ ævz. (we say rn approaches 0).

Czjw r ɯ 1g å (999/1000) DzU Pq U DUvz.

 

r<1 DzU Uuvg rAi CAv zU v Sn= a (1- rn) / (1-r) ====

Sinfinity = =

 

MAz Uuvg rAiİ S2n/ Sn = rn+1 JAz .

S2n/ Sn = [a(1- r2n)/(1-r)]/ [a(1- rn)/(1-r)]

= [a(1- r2n)*(1-r)]/[a (1- rn)*(1-r)]

= (1- r2n)/ (1- rn)

= (1- rn) (1+ rn)/ (1- rn) === (a2- b2) = (a-b)*(a+b) , r2n= (rn)2

= (1+ rn)

 

1.8.5 ĸ 2 : F jv tAi v PAqĻr: { 1,0.1,0.01,0.001,. (0.1)9} (UĤ: tAiİ 9 zU. 10zU.)

 

jg:

 

a=1, r=1/10

Sn = a (1- rn) / (1-r)

S10 = 1(1- (1/10)10 ) / (1-1/10)

= [(1010 -1)/1010]/(9/10)

= (1010 -1)/(9*109)

 

1.8.5 ĸ 3 : S10: S5= 33:1, T6= 32 Dzg D Uuvg rAiģ gɬj.

 

jg:

 

GPAiİ S10/ S5 = [a(r10-1)/(r-1)]/ [a(r5-1)/(r-1)]

= (r10-1)/ (r5-1)

= (r5+1) ===== UĤ: {(a2- b2) = (a-b)*(a+b) v r10= (r5)2}

 

Dzg S10/ S5 = 33 JAz Pnz.

(r5+1) =33

r5 =33-1=32 r =2

Tn = arn-1

T6 = a25

= 32(given)

a=1

{GP} = (1, 2, 4, 8, 16, 32,}

 

1.8.5 ĸ 4 : ê lֺ Pɮ U P̽U wAr AZĪ ƮP DZjPAz zsjwj.U AZĪU, 1 Pm Az DgA

w U Czg A U Pl PlU, 4 g PlU Pqwj. 341 wAr PlUzg, J UU ê wAr Pqz?

 

jg:

 

UU Aaz wAr PlU ASAi Uuvg rAiİz.

= {1,4,16,.} DU, a=1, r=4. Sn = 341 n=?

r >1Sn = [a(rn-1)/(r-1)]

Sn = a(4n-1)/(4-1)

= 1(4n-1)/3

= 341 zv

(4n-1) = 3Sn = 3*341=1023

4n= 1024

n =5

ê 5 UU wAr AZz

1.8.5 ĸ 5: M zAi 2 glP z ir, w A Plzg Jgqg Pqv zg, MAz wAU J z iqvۣ. ïwAi U ü. (ïw P 130)

 

jg:

 

z irz ASAi Uuvg rAiİz.

= {2,4,8,16,,.} DU, a=2, r=2, n=30

Sn = [a(rn-1)/(r-1)]

Sn = 2(230-1)/(2-1)

= 2(10243-1) ( 230 ={210}3=10243

= 2147483646

 

 

1.8.6 gvP r(Harmonic Progression):

 

F PɼV rU UĤ:-

{, , ,}

{,,}

ð rU zU êĪ gzU,

{ 3, 6, 9 12} Ez MAz iAvg r (ĸ1.8.3.3)

{8,18,28.} Ez MAz iAvg r (ĸ 1.8.3.1)

S:MAz rAi zU EAz iAvg rAi zU êUVzg, D rAiģ gvP r(Harmonic progression) Jߪg. Cz {HP} AiAv aĪg.

iAvg rAi{AP} i z = Tn =a+(n-1)d

{HP} Ai i z =

{HP}= {, , , . }

UĤ: gvP rAi zU v Sn PAqĻrAiĪ Aiiz v E.

1.8.6 ĸ 1 : MAz gvP rAiİ T4= , T10= Dzg T19 PAqĻrj.

jg:

gvP rAiİ Tn=

T4= = (zv)

T4= =

a+3d =12 ==========(1)

T10= = ( zv)

a+9d =42 ========== (2)

(1) (2) jAz PzU,

a+9d-(a+3d) =42-12

6d = 30

d =5

1 g d Ai ɯ 5 Dzò,

a+3*5 =12

a = (12-15) = -3

T19=

=

=

 

1.8.7 iAvg, Uuvg v gvP izsU(Arithmetic, Geometric and Harmonic means) (AM,GM and HM):

S: a, A v bU iAvg rAiĒ g zUzg, a v bU qī iAvg izs {Arithmetic Mean (AM)} A DVgvz.

a, A v bU iAvg rAi zU.

A-a =b-A( g Cx i v)

2A = a+b

A =

 

S: a, G v bU Uuvg rAiĒ g zUzg, a v bU qī Uuvg izs{Geometric Mean (GM)}G DVgvz.

a, G v b U Uuvg rAi zU.

=( i Cĥv)

G2= ab

G =

 

S: a, H v bU gvP rAiĒ g zUzg, H - a v bU qī gvP izs{Harmonic Mean (HM)}Vgvz.

a, H, b gvP rAi zUzg,

(,,) iAvg rAiiVgP.

DzjAz - = - ( i v)

= +

=

2ab =H(a+b)

H =

 

1.8.7 Ai: A, G v H U Jgq zs ASU qī iAvg izs (AM) Uuvg izs (GM) v gvP izs (HM)Uzg, A,G v H U Uuvg rAiİ JAz .

 

Pz: G/A =H/G (i Cĥv)

FU, (AM) A = (GM) G =

(HM) H =

 

A*H = * = ab= ()2= G2

Cx H/G = G/A DzjAz, A,G, H U Uuvg rAiİ.

UĤ: Aiiz Jgq zs ASUU, AGH DVgvz.( (a+b)2 v GAiV )

 

 

1.8 PPAi gA

 

 

P.A.

ɣqPz CAU

1

{AP}= {a, a+d, a+2d,a+3d ..a+(n-1)d} iAvg rAi i z: Tn= a+(n-1)d

2

=

3

iAvg rAiİ Sn = n*[2a+(n-1)*d]/2= n*(a+ Tn)/2

4

{GP} = {a, ar, ar2, ar3 .. ar(n-1)} Uuvg rAi i z:Tn= Tn-1*r = ar(n-1)

5

Uuvg rAi Sn = a(1- rn)/(1-r)

6

{HP}= {, , , } gvP rAi i z: Tn=

7

iAvg rAi iAvg izs (AM): A=

8

Uuvg rAi iAvg izs (GM): G =

9

gvP rAi iAvg izs (HM): H =