PROGRAM DIDIK CEMERLANG AKADEMIK

SPM

PROGRESSIONS (Geometric Progression)

ORGANISED BY:

JABATAN PELAJARAN NEGERI PULAU PINANG

ADDITIONAL MATHEMATICS

FORM 5MODULE 2

CHAPTER 2 : GEOMETRIC PROGRESSIONS

Contents Page

2.1 CONCEPT MAP (GEOMETRIC PROGRESSIONS)2

2.2 IDENTIFY CHARACTERISTICS OF GEOMETRIC PROGRESSIONS

2.3 DETERMINE WHETHER A GIVEN SEQUENCE IS A GEOMETRIC PROGRESSION

2.4 DETERMINE BY USING FORMULA

a) specific terms in geometric progressions b) The number of terms in geometric progressions

2.5 FIND

a) The sum of the first n terms of of geometric Progressions b) The sum of a specific number of consecutive terms of geometric Progressions c) The value of n ,given the sum of the first n terms of geometric Progressions

2.6 FIND a) The sum to infinty of geometric progressions

b) The first term or common ratio, given the sum to infinity

2.7 SOLVE PROBLEMS INVOLING GEOMETRIC PROGRESSIONS

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4

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SPM Questions 15

Assessment test18

Answers 20

CHAPTER 2 : GEOMETRIC PROGRESSION

2.1 CONCEPT MAP

1

PROGRESSION

PROGRESSIONS( Geometric Progression)

2

ARITHMETICPROGRESSION

GEOMETRIC PROGRESSION

GEOMETRIC PROGRESSION

Tn = arn-1

.a = first term

.r = Common ratio

.n = number of terms

SUM OF THE FIRST n TERMS

Sn = , r > 1

OR

Sn = , r < 1

THE n th TERM Tn

THE r th TERM Tr

T = S - S

SUM OF INFINITY

Sn = , -1 < r < 1

a = _______________r = _______________l = _______________n = _______________

2.2 Identify characteristics of geometric progression:

EXERCISE 1:Complete each of the sequence below when give a (fist term ) and r (common ratio) .

2.3 Determine whether a given sequence is an geometric progression

EXERCISE 2 :

2.3.1 Determine whether a given sequence below is an geometric progression.

Example : a) -8, 4, -2,……

.r = common ratio

.r = = , = (true)

b) 5, 11, 17, 23,…

A .r the first four terms of geometric progression,

Example:a) -3 2 -3, (-3)(2)1 = - 6, (-3)(2)2 = -12, (-3)(2)3 = -24

.b) 3 -2

.c) 4 3

.d) -6 -2

.e)

.f) y y

3

Fill in the blank

c) 16, -8, 4,…..

d) -20, -50, -30, -35,…..

e) x, x, x……

f) a5, a4 b, a3 b2

g) , , ,….

h) , , ,…

EXERCISE 3 :

2.3.2 Given that the first three terms of a geometric progression are below. Find the value of x

Geometric progression Value xExample : a) .x, x + 4, 2x + 2,… =

(x + 4)2 = x(2x + 2).x2 + 8x + 16 = 2x2 + 2x

0 = x2 – 6x – 190 = (x – 8)(x + 2).x – 8 = 0 @ x + 2 = 0

Hence x = 8 @ x = -2

b) x, x + 2 , x + 3

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c) x + 3, 5x - 3, 7x + 3

e) x – 6, x, 2x + 16

2.4 Determine by using formula:

EXERCISE 4:

2.4.1 specific terms in arithmetic progressions

Example :1. Find the 7 term of the geometric

progression. - 8, 4 , -2 , …..Solution:

a = - 8 r = =

2. Find the 8 term of the geometric progression.

16, -8, 4,…

5

Tn = arn-1

T7 = (-8)( )7-1

=

3. For the geometric progression

, , 1 , ….. ,find the 9 term .

4. Find the 3 term of the geometric progression 50, 40, 32…….

5. Find the 10 term of geometric progression a5, a4 b, a3 b2…………

6. Given that geometric progression 5.6, 1.4, 0.35, …… Find the 10th term.

2.4.2 Find the number of terms of the arithmetic progression

EXERCISE 5:

Example :

a) 64, - 32, 16,…….-

Tn = - a = 64, r = -

64 = -

=

=

=

b) 2, -4, 8, ……512

6

.n - 1 = 9 .n = 10

. c) 405, - 135, 45, …… - .d) , , 1…….. ,

2.5 Finda) The sum of the first n terms of geometric progressionsb) The sum of a specific number of consecutive terms of

geometric progressionsc) The value of n , given the sum of the first n terms of

geometric progressions.

2.5.1 Find the sum of the first n terms of geometric progressions

EXERCISE 6:

geometric progressions Find the sum of the first n term

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Sn =

1

)1(

r

ra n

, r > 1

OR

Sn = r

ra n

1

)1( , r < 1

Example :

.a) 2, - 4, 8,…………

.a = 2 r = = - 2

S7 = = 86

b) 5, 10, 20……….. S

c) 12, -6, 3……… S

d) x, x, x,…. S

2.5.2 The sum of a specific number of consecutive terms of geometric progressions

EXERCISE 7:

geometric progressions Find the sum of the first n term

Example 1

.a) 4, 2, 1,…………

.a = 4 r = =

Tn =

4 =

=

= 2

.n – 1 = 8 .n = 9

8

S = = 7

3 1, 3, 9,……. 2187.

c) 24, 12, 6, …….

2.5.3 The value of n , given the sum of the first n terms of geometric progressions.

EXERCISE 8 :

Example :

.a) The first and 4th tems of a geometric progression are and .

Find the value of rSolution :

T =

(r ) =

(r ) =

(r ) = , Hence r =

9

b) The first and 6th tems of a geometric progression are 2 and 607 .

Find the value of r

c) The common ratio and 5th tems of a geometric progression are and 7 .

Find the value of a

2.4 Find :a) the sum to infinity of geometric progressions

b) the first term or common ratio, given the sum to infinity of geometric progressions.

EXERCISE 9:

Find the sum to infinity of a given geomertric progression below:

Example:

1. 24, 3.6, 0.54, …….

10

a = 6

2. 81, -27,9, ……..

3. ..

EXERCISE 10:

1. The sum to infinity of a geometric progression is 200. Given that the first term is 52. Find the common ratio.

2. Given that the common ratio of

geometric progression is . The

sum of the first n terms,where n is large enough such that is 75. Findthe first term.

Example:The sum to infinity of a geometric

progression is 8. Given that the first term is 2. Find

a) the common ratiob) the third term

Solution: a)

3. The sum to infinity of a geometricprogression is 600 and the common ratio is 0.4 . Find

a) the first term b) the minimum number of

terms such that the sum of terms to be more then 599.

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b)

4. Express each of the recurring decimal below as a fraction in its simplest form.

Example:

0.3……

0.3…= 0.3 + 0.03 + 0.003 + …..

a) 0.444…….

Example:

4.020202……

c) 1.121212…..

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b) 0.232323…… d) 5.070707…...

2.5 Solve problems involving geometric progressions:

EXERCISE 11:

Example: A garderner has a task of cutting the grass of a lawn with an area of 1000 .

On the first day, he cut an area of . On each successive day, he cuts an area

1.1 times the area that he cut the previousday the task is completed. Find

a) the area that is cut on the 10th day.b) The number of the days needed to

complete the task.

Solution:a) a=16 , r=1.1

Osman is allowed to spend an allocation of RM1 million where

the maximum withdrawal each day must not exceed twice the amount withdrawn the day before. If Osman withdraws RM200 on the first day, determine after how many days the amount of money allocated will all be used up.

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SPM QUESTIONS:

1. 2003 (Paper 1: No.8)

In a geometric progression, the first term is 64 and the fourth term is 27. Caculate(a) the common ratio(b) the sum to infinity of the geometric progression. [4 marks]

2. 2004(Paper 1: No.9)

Given a geometric progression ,express p in terms of y.

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3. 2004(Paper 1: No.12 )

Express the recurring decimal 0.969696……as a fraction in its simplest form. [4 marks]

4. 2004(Paper2: Section A: No.6)

Diagram 2 shows the arrangement of the first three of an infinite series of similartriangles. The first triangle has a base of x cm and a height of y cm. The measurements of the base and height of each subsequent triangle are half ofthe measurements of its previous one.

Diagram 2

(a) Show that the areas of the triangles form a geometric progression and statethe common ratio. [3 marks]

(b) Given that x= 80 cm and y= 40 cm,

(i) determine which triangle has an area of ,

(ii) find the sum to infinity of the areas, in , of the triangles. [5 marks]

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y cm

xcm

5. 2005 (Paper 1 : No.10)

The first three terms of a sequence are 2 , x , 8Find the positive value of x so that the sequence is(a) an arithmetic progression(b) a geometric progression [4 marks]

6. 2005 (Paper 1: No. 12)

The sum of the first n terms of the geometric progression 8,24,72,….is 8744.Find(a) the common ratio of the progression(b) the value of n [4 marks]

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ASSESSMENT:

1. The first three terms of a geometric progression are 2x + 3, x and x – 2 with a common ratio r , where -1 < r < 1. Find

(a) the value of x (b) the sum of the first n terms ,where n is large enough such

that

2. In the progression 5 , 10 , 20 , 40 , ……. Find the least number ofterms required such that their sum exceeds 1000.

3. The third term and the sixth term of a geometric progression are27 and 8 respectively. Find the second term.

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4. In a geometric progression, the sum of the first five terms is .

Given that the common ratio is . Find

(a) the first term(b) the sum of all the terms from the fourth to the sixth term.

5. The third term of a geometric progression exceeds the second termby 6 while the fourth term exceeds the third term by 2. Find thesum of the first 5 terms.

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ANSWERS:EXERCISE 1:

b) 3, 6, -12, 24c) 4, 12, 36, 108d) -6, 12, 124, 48

e) , , ,

f) , , ,

EXERCISE 2:a) trueb) falsec) trued) falsee) truef) trueg) falseh) true

EXERCISE 3:b) x = -4c) x = 3d) x = -12 @ x = 8

EXERCISE 4:

1. T = -

2. T = -

3. T =

4. T = 32

5. T =

6. T = 0.000021

EXERCISE 5:b) n = 9c) n = 8d) n = 7

EXERCISE 6:b) 275

c) 8

d)

EXERCISE 7:c) n = 8, S = 3280

d) n = 9, S = -1022

e) n = 6, S = 47

EXERCISE 8:b) r = 3c)a = 3

EXERCISE 9: 1. 28.24

EXERCISE 10:2. r=0.74 3. a=72

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2.

3. 1

4. a) 360 b) 7

5.

EXERCISE 11: 13 days

SPM QUESTION:

1. a) b)

2.

3.

4. a)

b)i. n=5 ii.

5. a) x=5 b) x=4 6. a) r=3 b) n=7

ASSESSMENT:

1. a) x=3 b)

2. 8

3.

4. a) -2 b)

5.

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