progression.doc

8/12/2019 progression.doc

1/25

PROGRESSION

1. ARITHMETIC PROGRESSION

1.1 Identify characteristics of arithmetic proression ! AP"

E#amp$e 1 % e#amp$es of AP

& ' ( ' ) ' 11 ' ****

1& ' 11 ' ( ' + ' ***..

1., -etermine hether a i/en se0ence is an AP

E#amp$e , % -etermine hether each of the fo$$oin se0ences is an AP

a. 2 ' ( ' 13 ' 1+ ' 14 ' **.

It is an AP

5. 1 ' 2 ' ) ' 14

It is not an AP

E#ercise , % -etermine hether each of the fo$$oin se0ences is an AP

a. + ' & ' ( ' ) ' *..5. ......'

6

1'

2

1'

,

1'1

c. 13 ' ( ' 2 ' 1 ' *.. d. ,a ' ,a 7 b ' ,a7 ,b' ,a7 +b' *..

1

7, 7 , 7 ,

8 2 8 2 8 2

7 + 7 + 7 + 7 +7 + 7 &

AP is a se0ence of nm5ers here the difference 5eteen sccessi/e terms is a constant.

If a se0ence is an AP ' then the difference 5eteen sccessi/e terms is a constant.


2/25

1.+ -etermine 5y sin form$a

a. specific terms in AP

5. the nm5er of terms in AP

E#amp$e + %

a. 9ind the 6thterm of the AP

2 ' ( ' 13 ' 1+ ' 14 ' *.

a: 2 ' d: +

Tn: a7 ! n; 1 " d

T6: 2 7 ! 681" + : ,&

5. 9ind the 1,thterm of the AP

11 ' ( ' + ' 81 ' *

a: 11 ' d: 82

Tn: a7 ! n; 1 " d

T1,: 11 7 ! 1,81"!82" : 8 ++

c. 9ind the nthterm of the fo$$oin AP

+ ' 6 ' 1+ ' 16 ' *

a: + ' d: &

Tn: a7 ! n; 1 " d : + 7 ! n81" &

: + 7 &n8 &

: &n ; ,

d. 9ind the nm5er of terms in the AP

1& ' 16 ' ,1 ' * ' 2,

a: 1& ' d: + ' Tn: 2,

Tn: a7 ! n; 1 " d 2, : 1& 7 ! n81" +

2, : 1& 7 +n8 +

n: 13

,

An AP can 5e represented as

a , a 7 d , a + 2d , a + 3d , a + 4d ,

In enera$ ' the nthterm ' T nis represented 5y

T n: a7 ! n; 1 " dhere a; first term

d; common difference n; nm5er of terms

9irst term '

a

Second term ' T,

Third term ' T+

9orth term ' T2

9ifth term ' T&


3/25

E#ercise +.

1. 9ind the 1,thterm of the fo$$oin AP

a. 2 ' ( ' 13 ' 1+ ' *. 5. 11 ' ( ' + ' 81 ' *

c. ,6 ' ,2 ' ,3 ' *d. ....'

,

1&'4'

,

)'+

,. 9ind the nthterm of the fo$$oin AP

a. 4 ' 1+ ' ,3 ' ,( ' *. 5. 16 ' 1& ' 1, ' *.

c. 8,3 ' 814 ' 81, 'd. ....'

2

1&'+'

2

)'

,

+

+. 9ind the nm5er of terms in the fo$$oin AP

a. &3 ' 2( ' 22 ' **8+2 5. 6 ' 11 ' 12 ' *.' &)

c. 8,3 ' 814 ' 81, ' *14 d.2+)'....'

21&'+'

2)'

,+

+


4/25

2. Gi/en that 4x; , ' +xand ,x7 2 are the first three terms of an AP. 9ind

a. the /a$e ofx'5. the se/enth term of the AP.

&. The fifth term of an AP is +2 and the e$e/enth term is &,. 9ind the fifteenth term.

4. The eihth term and the eihteenth term of an AP is +( and )( respecti/e$y. 9ind

a. the first term and the common difference 5. the tenth term.

2


5/25

(. Gi/en the AP ' 812 ' 8& ' 2 ' 1+ ' ** 9ind the term hich is e0a$ to (4.

6. In an AP' the nthterm ' Tnis i/en as Tn: (n; +. 9ind

a. the first term

5. the common difference c. the 16thterm.

&


6/25

1.2 9ind%

a" the sm of the first nterms of arithmetic proressions.

5" the sm of a specific nm5er of consecti/e terms of arithmetic proressions.

c" the /a$e of n' i/en the sm of the first nterms of arithmetic proressions.

E#amp$e 2

a. 9ind the sm of the first ten terms ofthe AP 2 ' 4 ' 6 ' 13 ' *.

a: 2 ' d: ,

( ){ }

( ) ( ){ }

113

,162,,

13

1,,

=

+=

+= dnan

Sn

5. 9ind the sm of the first 16 terms ofthe AP ; 11 ' 86 ' 8& ' *

a: 811 ' d: +

( ){ }

( ) ( ){ }

,41

+11611,,

16

1,,

=

+=

+= dnan

Sn

c. 9ind the sm of the AP d. The sm of the first nterms of an AP

4

1. The sm of the first nterms of an AP ' Snis i/en 5y

i. ( ){ }dnan

Sn

1,,

+=

ii. { }lan

Sn

+=,

herel is the $ast term < l: Tn: a7 !n81"d=

,. To determine the nthterm of any proression '

1= nnn SST


7/25

&3 ' 24 ' 2, ' *.'12

9ind nfirst

a : &3 ' d: 82 ' l: Tn: 12

Tn: a7 ! n; 1 " d 12 : &3 7 ! n; 1 "82

8+4 : 82n7 2 2n: 23

n: 13

( ){ }

( ) ( ){ }

+,3

2113&3,,

13

1,,

13

=

+=

+=

S

dnan

Sn

( ' 13 ' 1+ ' 14 ' *. is +(1. 9ind the

/a$e of n.

( ){ }

( ) ( ){ }

{ }

12

444(.1('12

"+!,

"(2,"!+!2"11!11

3(2,11+

+11(2,

++12(2,

+1(,,

+(1

1,,

,

,

,

==

=

=+

+=

+=+=

+=

n

n

n

nn

nn

nn

n

n

dnan

Sn

E#ercise 2

a. 9ind the sm of the first 1, terms of

the AP + ' 4 ' ) *.

5. 9ind the sm of the first 13 terms of

the AP 84 ' 811 ' 814 ' *.

c. Gi/en the first three terms of an AP is( ' 13 ' 1+ ' *9ind the sm of the first

4 terms after the fifth term.

d. Gi/en the sm of the first nthterms ofan AP ( ' 1+ ' 1) ' *.. is ,,2. 9ind the

/a$e of n.

e. Gi/en that the sm of the first n terms ' Snof an AP is i/en 5y Sn: ,n7 n,.

9ind

(


8/25

i. the first term

ii. the common difference

f. In an AP ith 1& terms ' the forth term is 16 and the tenth term is 2,. 9ind the sm

of a$$ the terms in the proression.

1.& So$/e pro5$ems in/o$/in AP

1. In an AP ' the fifth term is 14 and the sm of the first eiht terms is 114. 9ind thesm of a$$ the terms from the si#th term to the te$fth term.

,. Gi/en the first three terms of an AP are &x84 ' 14 ;xand +x; ,. 9ind the /a$e ofx

and the sm of the first 1, terms.

6


9/25

+. The sm of the first 6 terms of an AP is ,6 and the sm of the ne#t 6 terms is 8142 . 9ind the

a. first term and the common difference

5. sm of a$$ terms from the 13th

term to the ,3th

term.

2. Gi/en an AP 8& ' 8, ' 1' *. State the for consecti/e terms in this proression thatsms p to &6.

,. GEOMETRIC PROGRESSION

)


10/25

,.1 Identify characteristics of eometric proression ! GP"

8 E#amp$e 1 % e#amp$es of GP

, ' 6 ' +, ' 1,6 ' ****

, ',

1 '

6

1' ..

,., -etermine hether a i/en se0ence is a GP

E#amp$e , % -etermine hether each of the fo$$oin se0ences is a GP

a. 1 ' + ' ) ' ,( ' **.

It is a GP

5. ...'14

1'

2

1'

,

1

It is not a GP

E#ercise , % -etermine hether each of the fo$$oin se0ences is a GPa. 2 ' 81,' +4 ' 8136 ' *..

5. ......1),

1'

26

1'

1,

1'

+

1

c. 1 ' 2 ' ) ' 14' *.. d. + ' ) ' ,( ' 61 *..

,.+ -etermine 5y sin form$a

c. specific terms in GPd. the nm5er of terms in GP

13

# 2 # 2 # 2

2

1

1

# + # + # +

,

1

2

1

GP is a proression in hich each term is m$tip$ied 5y a constant in order to o5tain the

ne#t term.

2

1


11/25

E#amp$e + %

a. 9ind the eihth term of the GP

+ ' 4 ' 1, ' *.

a: + ' r: ,

Tn: arn81

T6: !+"!,"681

: +62

5. 9ind the si#th term of the AP

....'1,

1'

4

1'

+

1

a:+

1' r:

,

1

Tn: arn8 1

T4:14

,1

+1

:)4

1

c. 9ind the nthterm of the fo$$oin GP

+ ' ) ' ,( *

a: + ' r: +

Tn: arn81

Tn: !+"!+"n81 : + 17!n81"

: +n

d. 9ind the nm5er of terms in the GP

+ ' 84 ' 1, ' * ' 81&+4

a: + ' r: 8, '. Tn: 81&+4

Tn: arn8 1

81&+4 : !+"!8,"n81

8&1, : !8,"n81 !8,"): !8,"n81 ) : n; 1

n : 13

E#ercise +.

1. 9ind the (thterm of the fo$$oin AP

a. + ' 4 ' 1, ' *. 5. , ' 84 ' 16 ' *

11

An GP can 5e represented as

a , ar , ar, , ar3 , ar4 ,

In enera$ ' the nthterm ' T nis represented 5y

T n: arn; 1 here a; first term

r; common ratio

n; nm5er of terms

9irst term 'a

Second term ' T,

Third term ' T+

9orth term ' T2

9ifth term ' T&


12/25

c. , ' 2 ' 6 ' *d. ....'

14

1'

6

1'

2

1

,. 9ind the nthterm of the fo$$oin AP

a. , ' 4 ' 16 ' *. 5. 1, ' 4 ' + ' *.

c. + ' 4 ' 1, ' *.d. ....','

,

1'

6

1

+. 9ind the nm5er of terms in the fo$$oin AP

a. , ' 4 ' 16 ' *. 264

5. +,3 ' 143 ' 63 ' *,3

c.13,2

1...''

6

1'

2

1'

,

1d.

,2+

1...''

+

1'1'+

2. Gi/en that 2 'xand ) are the first three terms of a GP. 9ind

a. the positi/e /a$e ofx'

5. the se/enth term of the GP.

1,


13/25

&. The second term and the forth term of a GP is 2 and +4 respecti/e$y.Gi/en that the

common ratio is positi/e. 9inda. the common ratio

5. the si#th term.

6. 4. Gi/en the GP ...')'+'1'

+

1 9ind the term hich is e0a$ to (,).

,.2 9ind%

a" the sm of the first nterms of eometric proressions>

5" the sm of a specific nm5er of consecti/e terms of eometric proressions.

c" the /a$e of n' i/en the sm of the first n terms of eometric proressions.

1+


14/25

E#amp$e 2

a. 9ind the sm of the first se/en terms of

the GP 2 ' 6 ' 14 ' *.

a: 2 ' r : ,

( )

( )[ ]

&36

1,

1,2

1

1

(

(

=

=

=

S

r

raS

n

n

5. 9ind the sm of the first & terms of the

GP 26 ' 81, ' + ' *..

a: 26 ' r:

2

1

14

41&

2

11

2

1126

1

1

&

&

=

=

=

S

r

raS

n

n

c. 9ind the sm of the GP& ' 813 ' ,3 ' *'+,3

9ind nfirst

a : & ' r: 8, ' l: Tn: +,3

d. The sm of the first nterms of a GP1 ' , ' 2 ' 6 *. is ,&&. 9ind the

/a$e of n.

12

1. The sm of the first nterms of a GP ' Snis i/en 5y

i.( )

r

raS

n

n

=

1

1 , r 1

ii.( )

1

1

=

r

raS

n

n , r 1

,. To determine the nthterm of any proression '

1= nnn SST


15/25

(

14

",!",!

",!42

","!&!+,3

14

1

1

1

=

=

=

=

=

=

n

n

arT

n

n

n

n

n

( )

,1&

",!1

,1& (

(

=

=S

( )

( )[ ]

6

,,

,,&4

1,

1,1,&&

1

1

6

==

==

=

n

r

raS

n

n

n

n

n

E#ercise 2

a. 9ind the sm of the first 1, terms ofthe GP + ' 4 ' 1, ' *.

5. 9ind the sm of the first 13 terms ofthe GP 42 ' 814 ' 2 ' 81 ' *.

c. Gi/en the first three terms of a GP is

+ ' ) ' ,( ' * 9ind the sm of the first

4 terms after the forth term.

d. Gi/en the sm of the first nterms of

a GP 2 ' 6 ' 14 ' *.. is &36. 9ind the

/a$e of n.

e. Gi/en that the sm of the first nterms ' Snof a GP is i/en 5y Sn: 2! ,n; 1 ". 9ind

i. the first term

ii. the common ratio.

1&


16/25

f. Gi/en that the third term of a GP is ,3 and the eihth term is 423. 9ind the sm ofthe first ten terms.

,.& 9ind%a" the sm to infinity of eometric proressions.

5" the first term or common ratio' i/en the sm to infinity of eometric proressions.

14

The sm to infinity of a GP hen 81 ? r ? 1 is i/en 5y

r

aS

1


17/25

E#amp$e &

1 . 9ind the sm to infinity for the fo$$oin GP

a. 14 ' 6 ' 2 '*.

a: 14 ' r :,

1

+,

,11

14

1

=

=

r

aS

5. ,( ' 8) ' + ' *.

a: ,( ' r :+

1

2

61

+11

,(

1

=

=

r

aS

,. E#press the recrrin decima$ as a fraction in its $oest term

a. 3.&&&**

3.&&& *..: 3.& 7 3.3& 7 3.33& 7 *

a: 3.& ' r : 1.3&.3

3&.3=

)

&

1.31

&.3...&&&.3

=

=

5. 3.+4+4+4*

3.+4+4+4: 3.+4 7 3.33+4 7 3.3333+4 7 *..

a: 3.+4 ' r : 31.3+4.3

33+4.3=

11

2

31.31

+4.3...+4+4+4.3

=

=

E#ercise &

1. 9ind the sm to infinity for the GP

...'+,

1'

6

1'

,

1

,. 9ind the sm to infinity for the GP

,( ' ) ' + ' *.

1(


18/25

+. 9ind the sm to infinity for the GP

...'2

,(')'1,

2. E#press the recrrin decima$3.&2&2* as a fraction in its $oest

term

&. E#press the recrrin decima$

3.,),),) * as a fraction in its $oestterm

4. E#press the recrrin decima$

3.(((* as a fraction in its $oestterm

(. A GP is i/en 4 ' + ',

+' * 9ind the sm to infinity of this proression

16


19/25

6. In a GP ' the first term is 1,& and the forth term is 6. 9ind

a. the common ratio 5. the sm to infinity of this proression.

@. Gi/en that the sm to infinity of a GP is 23 and the first term is 13. 9ind the

common ratio

,.4 So$/e pro5$ems in/o$/in GP

1. The second term of a GP is 42 and the sm of the first to terms is 1),. 9ind the i. common ratio and the first term ii. the sm of a$$ the terms from the forth term to the eihth term.

1)


20/25

,. 9ind the $east nm5er of terms in the GP 2 ' 1, ' +4 ' *. that is reater than 4333.

+. In a GP ' the fifth term is2

61and the sm of the third term and forth term is ) .

Gi/en that r ? 3 ' find

ii. the common ratioiii. the first termi/. the sm of the first 4 terms.

,3


21/25

PAST EAR BESTIONS

Paper 1

1. It is i/en that the first for terms of a GP are + ' 84 ' 1, andx. 9ind the /a$e ofx.

,. The first three terms of an AP are 24 ' 2+ and 23. The nth term of this proression is

neati/e. 9ind the $east /a$e of n.

+. a" -etermine hether the fo$$oin se0ence is an AP or a GP

14x' 6x' 2x' *

,1


22/25

5" Gi/e the reason for the ansers in )!a".

2. Three consecti/e terms of an AP are & ;x' 6 ' ,x. 9ind the common difference of

the proression.

&. The first three terms of a GP are ,( ' 16 ' 1,. 9ind the sm to infinity of the GP.

4. The ninth term of an AP is 2 7 &p and the sm to infinity of the first for terms ofthe proression is (p; 13' herep is a constant. Gi/en that the common difference

of the proression is &' find the /a$e ofp.

(. The third term of a GP is 14. The sm of the third term and the forth term is 6.9ind

a. the first term and the common ratio of the proression

5. the sm to infinity of the proression.

6. The first three terms of a se0ence are , 'x' 6. 9ind the positi/e /a$e ofxso that

the se0ence isa. an arithmetic proression

5. a eometric proression

). The first three terms of an arithmetic proression are & ' ) ' 1+. 9ind

a. the common difference of the proression

5. the sm of the first ,3 terms after the third term.

13. The sm of the first nterms of a GP 6 ' ,2 ' (, ' * is 6(22. 9ind

a. the common ratio of the proression

5. the /a$e of n.

11. Gi/en a GP ...'&

2',' py ' e#presspin terms ofy.

1,. Gi/en an AP 8( ' 8+ ' 1 ' * ' state three consecti/e terms in this proression hichsm p to (&.

1+. E#press the recrrin decima$ 3.)4)4)4* as a fraction in its simp$est form.

12. The first three terms of an arithmetic proression are k; + ' k7 + ' ,k7 ,. 9ind

a. the /a$e of D 5. the sm of the first nine terms of this proression.

1&. In a GP' the first term is 42 and the forth term is ,(. Ca$c$ate

a. the common ratio5. the sm to infinity of the GP

Paper ,

,,


23/25

1. Mth started orDin for a company on 1 anary ,33, ith an initia$ anna$

sa$ary of RM16 333. E/ery anary' the company increased his sa$ary 5y &F of thepre/ios years sa$ary. Ca$c$ate

a. his anna$ sa$ary ' to the nearest RM ' for the year ,33(

5. the minimm /a$e of n sch that his anna$ sa$ary in the nth year i$$ e#ceedRM+4 333.

c. the tota$ sa$ary ' to the nearest RM' paid to him 5y the company ' for the years

,33, to ,33(.

,. -iaram 1 shos the side e$e/ation of part of stairs 5i$t of cement 5$ocDs.

The thicDness of each 5$ocD is 1& cm. The $enth of the first 5$ocD is )6& cm. The$enth of each s5se0ent 5$ocD is +3 cm $ess than the precedin 5$ocD as shon in

diaram ,.

a. If the heiht of the stairs to 5e 5i$t is + m' ca$c$ate i. the $enth of the top most 5$ocD

ii. the tota$ $enth of the 5$ocDs

5. Ca$c$ate the ma#imm heiht of the stairs.

+. To companies ' -e$ta and Omea ' start to se$$ cars at the same time.

a. -e$ta se$$s kcars in the first month and its sa$e increases constant$y 5y mcarse/ery s5se0ent month. It se$$s ,23 cars in the 6 thmonth and the tota$ sa$es for

the first 13 months are 1)33 cars.

9ind the /a$e of kand of m.

,+

1& cm

),& cm

)&& cm

)6& cm

-iaram 1


24/25

5. Omea se$$s 63 cars in the first month and its sa$es increase constant$y 5y ,,

cars e/ery s5se0ent month.

If 5oth companies se$$ the same nm5er of cars in the nth month' find the /a$eof n.

2. -iaram , shos part of an arranement of 5ricDs of e0a$ sie.

The nm5er of 5ricDs in the $oest ro is 133. 9or each of the other ros' thenm5er of 5ricDs is , $ess than in the ro 5e$o. The heiht of each 5ricD is 4 cm.

A$i 5i$ds a a$$ 5y arranin 5ricDs in this ay. The nm5er of 5ricDs in thehihest ro is 2. Ca$c$ate

a. the heiht ' in cm ' of the a$$

5. the tota$ price of the 5ricDs sed if the price of one 5ricD is 23 sen.

&. -iaram + shos the arranement of the first three of an infinite series of simi$ar

trian$es. The first trian$e has a 5ase ofxcm and a heiht ofycm. The

measrements of the 5ase and heiht of each s5se0ent trian$e are ha$f of themeasrements of its pre/ios one.

a. Sho that the areas of the trian$es form a eometric proression and state thecommon ratio.

5. Gi/en that # : 63 cm and y : 23 cm'

i. -etermine hich trian$e has an area of2

14 cm,.

ii. find the sm to infinity of the areas' in cm,' of the trian$es.

ANSERS

,2

4 cm

-iaram ,

ycm

xcm

-iaram +


25/25

Paper 1

1. 8,2 ,. 1(

+. a. GP 5. the ratio of to consecti/e terms of the se0ence is a constant.

2. 12 &. 61 4. 6 (. a a : 42,

1=r 5.

+

,2,

6. a. AP 5. GP ). a. 2 5. 1133 13. a. + 5. (

11. ,6

y

p = 1,. ,1 ' ,&' ,) 1+.++

+,

12. a. ( 5. ,&,

1&. a.2

+

5. ,&4

Paper ,

1. a. RM ,, )(+ 5. n : 14 c. RM 1,, 2+2

,. a. i. 21& cm ii. 12 333 cm 5. 2)& cm

+. a. m : ,3 ' D : 133 5. 11

2. a. ,)2 cm 5. RM 131).,3

&. a. Trian$es form a GP ith common ratio :2

1

5. i. &thtrian$e ii.+

1,1++

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