1. progressions

7/31/2019 1. Progressions

1/25

1

SM SAINS SERI PUTERI, KUALA LUMPUR

PROGRESSIONS

NAME:

CLASS: .


2/25

1.1 Characteristics of Arithmetic Progression

A sequence with a common difference d is called an arithmetic progression.

If T1, T2, T3, T4, Tn, are the terms of an arithmetic progression,

Then

a, the first term =

d, the common difference = T2 T1 = T3 T2 = Tn Tn-1

Exercise 1.1

Determine the first term and the common difference of this arithmetic progression.

1. 0.3, 1.0, 1.7, 2.4,

a = _____________ d= __________________ = _____________

2. ,...

8

10,

8

7,

8

4,

8

1

a = _____________ d= __________________ = ______________

1.2 To Determine Whether A Sequence Is An Arithmetic Progression

Determine whether each of the following is an arithmetic progression:

1. 1, 2, 5, 8,

T2 T1 = ______________

T3 T2= ______________

T4 T3= _______________

2


3/25

Common difference, d= ___________

The sequence is an ____________________________

2. 13y, 16y, 20y, 23y,

T2 T1 = ______________

T3 T2= ______________

T4 T3= _______________

There is no common difference

The sequence is ____________________________

Exercise 1.2

Refer to Test Book Page 3, Skill Practice 2.

Enrichment Exercise

1. k+3 , 2k+ 6 ,8 are the first three terms of an arithmetic progression, find the

value ofk. (Answer: 13

k= )

2. Given that2,5 ,7 4x x x are three consecutive terms of an arithmetic progression

Wherex has a positive value. Find the value ofx. (Answer:x = 4)

3

Note:

Ifx,y andzare three terms of anarithmetic progression ,

y x = z - y


4/25

3. Given that the first three terms of an arithmetic progression are 2 ,3 3 and 5y+1y y + .

Find the value ofy. (Answer: y = 5)

4. Diagram 4(i) shows three squares with sides increase 1 unit successively. Diagram

4(ii) shows three rectangles with constant width and the length increase 1 unit

successively.

Show that areas of the rectangles in Diagram 4(ii) forms an arithmetic progression,whereas areas of the squares in Diagram 4 (i) is not.

x x + 1 x + 2Diagram 4(i)

p p p x + 2q q + 1 q + 2

Diagram 4(ii)

4


5/25

1.3 The nth term of an arithmetic progression

Tn = a + (n 1)d

Exercise:

1. Skill Practice 3 (page 5)

2. If the common difference of an arithmetic progression is 7 and the 6th term is 38.

Find:

(a) the first term, a,(b) the 25th term.

1.4 The number of terms in an arithmetic progression

Exercise:


2. For the following arithmetic progression: 13, 8, 3, 2, , 72

Find:

(a) the 15th term,(b) the number of terms in the sequence.

5


6/25

3. Find the 5th and the 50th term in the following arithmetic progression if

given below are the 16th, 17th, and 18th terms:

15, 24, 33,

Find the number of terms in the following arithmetic progression:

2r+ 3s, r+ 5s, 7s, r+ 9s, , 8r+ 23s

6


7/25

1.5 The sum of the first n terms of an arithmetic progression

])1(2[2

][2

dna

n

lan

Sn

+=

+=

Examples

1. Find the sum of the first 10 terms of the arithmetic progression 2, 6, 10,


2. (a) Find the sum of the following arithmetic progression 4, 8, 12, , 60.

7


8/25

Enrichment Exercise:

1. SPM 2003: Paper 1

The first three terms of an arithmetic progression are 3, 3,2 2k k k + +

Find

(a) the value ofk,(b) the sum of the first 9 terms of the progression. [ ]3 marks

[Answers: (a) 7 (b) 252]


Given an arithmetic progression 7, 3, 1, . , state three consecutive terms in this

progression which sum up to 75. [ ]3 marks

[Answer: 21, 25, 29]

8


9/25


The volume of water in a tank is 450 litres on the first day. Subsequently, 10 litres ofwater is added to the tank everyday.

Calculate the volume, in litres, of water in the tank at the end of the 7 th day.

[ ]2 marks [Answer: 510]

9


10/25

1.6 Finding the number of terms in a given sum.

Examples:

1. How many terms will add up to the sum of 252 for an arithmetic progression

7, 14, 21, ?

2. Given that the sum of the first n terms of the arithmetic progression 15, 23, 31,

is 708, find the value ofn.

10


11/25

1.7 Sum of the specific consecutive terms of an arithmetic progression

1. Given andarithmetic progression: T1, T2, T3, T4, Tn,S3 = T1 + T2 + T3

S4 =

S5 =

S4 S3 =

S5 S4 =

Hence, Tn =

2. S10 =

The sum from the 3rd term to 5th term = ST3to T5

= T3 + T4 + T5

= S5 S2

The sum from the 4rd term to 10th term = ST4to T10= T4 + T5 + T6 + T7 + T8 + T9 + T10

= S10 S3

Hence, STa to Tb = Sb Sa-1

11


12/25

Examples:

1. Find the sum of all the terms from the seventh term to the eleventh term of thefollowing arithmetic progression: 8, 13, 18,

2. Given that the sum of the first n terms of an arithmetic progression is 2n2 + 3n,find the twelfth term.

3. The 4th term of an arithmetic progression is 2 and the 10th term is 16. Find

the sum from the 5th term to the 16th term.

12


13/25

1.8 Solve Problems Involving Arithmetic Progression

Polyas Problem-Solving Steps:

1. Understand the Problem

2. Plan the Strategy

3. Implement the Strategy

4. Check the Answer

Examples:

1. When Raju gets his salary for the first month of working, he spends RM750 on general

expenses. The following month, he spends RM120 lesser from the 1st month. This

pattern continues until the 7

th

month.Find:

(a) Rajus expenses for the 5th month.(b) The total expenses of the first 7 months.

2. Diagram shows three circles which are drawn continuously on a horizontal lineMN.

The radius of each circle increases by 2 cm as compared to the one before. Suppose thepattern continues and more circles are drawn.

Find:

(a) the radius of the 7th circle.

(b) The total length of the perimeter of the 3rd circle to the 7th circle in terms of.

13

3 cm 5 cm

7 cm

M N


14/25

2. GEOMETRIC PROGRESSIONS

2.1 Characteristics of Geometric Progressions

A sequence with a common ratio, r is called a geometric

progression.

If T1, T2, T3, T4, Tn, are the terms of a geometric progression,

Then

a, the first term =

r, the common ratio = ==2

3

1

2

T

T

T

T

Exercise 2.1Determine the first term and the common ratio of this geometric progression:

(a) 8, 24, 72,

a = r= =

(b) 4, 20, 100,

a = r= =

(c) 2x, 8x, 32x,

a = r= =

14


15/25

2.2 To Determine Whether A Sequence Is A Geometric Progression

Determine whether each of the following is a geometric progression:

(a) 2, 12, 72, 432,

=

=

=

3

4

2

3

1

2

T

T

T

T

T

T

Common ratio =

The sequence is _____________________________

(b) 2, 4, 12, 24,

==

==

==

3

4

2

3

1

2

T

T

T

T

T

T

There is no _________________________________

The sequence is _____________________________

3. Skill practice 10 (Page 14)

15


16/25

2.3 The nth term of a geometric progression

Tn =

Examples:

1. Given a geometric progression:

5, 10, 20, 81920

Find :

(a) The expression for the nth term [ 5(2n-1) ]

(b) The seventh term [ 320 ]

(c) The number of terms in the progression [ 15 ]

16


17/25

2. For the following geometric progression:

13, 39, 117, , 255879

Find

(a) The 9th term [85293]

(b) The number of terms in the sequence. [10]

3. Given thatx + 2,x + 3,x + 6 are the first three terms of a geometric progression.Find:

(a) the value of x

(b) the value of the common ratio, r.

17


18/25

2.4 Sum of Geometric Progression

The sum of the first n terms of a geometric progression

Sn = , r > 1

Sn = , r < 1

Examples:

1. Find the sum of the first seven terms of the geometric progression

4, 8, 16, 32,

2. Find the sum of the first ten terms of the geometric progression

32, 16, 8, 4,

18


19/25

3. Find the sum of the following geometric progression

3, 9, 27, ..., 729

4. Find the sum from the sixth term to the twelfth term of the geometric progression 21,

42, 84,

5. The sum of the geometric progression 10, 16, 25.6, is 11, 993. Find the number of

terms in this progression.

19


20/25

2.5 Sum To Infinity Of A Geometric Progression

S =

Examples:

1. A geometric progression has the first term of 45 and the common ratio of .2

1What

is the sum to infinity of the geometric progression?

2. Find the sum to infinity of the following geometric progression:

2.5, 1.75, 1.225,

3. The sum to infinity of a geometric progression is 5. If the first term is 1.5, find the

common ratio of the progression.

20


21/25

2.6 Express A Recurring Decimal As A Fraction

Recurring Decimals:

Examples

1..

6.0 =

2. Write the recurring decimal..

27.0 as the sum of a geometric progression.

Hence find the sum to infinity of the progression.

4. Skill Practice 18 (Page 21) : No. (c), (e), (f)

21


22/25

2.6 Solving Problems Involving Geometric Progressions

Examples:

1. In a geometric progression the 3rd term is more than the 2nd term by 45

and the 2nd

term is more than the 1st

term by 30. Determine(a) the first term and the common ratio of this progression.

(b) the minimum number of terms such that its sum exceeds 2000.

2. If p + 1,p 7 andp 13 are the first three terms of a geometric

progression, find

(a) the value ofp,(b) the sum to infinity of the progression.

22


23/25

PRACTICE MAKES PERFECT

1. Show that log h, log hk, log 2hk , log 3hk , is an arithmetic progression. Then

find the common difference of this progression.

2. An arithmetic progression has 10 terms. The sum of all these 10 terms is 220. Thesum of the odd terms is 100. Find the first term and the common difference.

Answer : 4,4 == da

3. The sum of the first six terms of an arithmetic progression is 120. The sum of thefirst six terms is 90 more than the fourth term. Calculate the first term and the common

difference.

Answer : 20,30 == da

4. Given that the sum ofn term of an arithmetic progression is .322 nnSn += Find

(a) the n term in terms ofn

(b) the first term

(c) the common difference

Answer : (a) 14 +n (b) 5( c) 4

5. An arithmetic progression has 12 terms. The sum of all these 12 terms is 222. The

sum of the odd terms is 102. Find

(a) the first term and the common difference(b) the last term

Answer :(a) a = 2 , d = 3 (b) 35

6. The n term of an arithmetic progression is 85 n . Find the sum of all the terms

from the 5th term to the 8th term.Answer : 98

7. Estimate the sum to infinity of the geometric progression .......3

1139 ++++

Answer :

2

113

8. Write ........007.0007.007.07.0 ++++ as a fraction.

Answer :9

7

9. The sum of the first n terms of a geometric progression is3

8)2( 12 =

+n

nS . Find

the least number of terms in the progression that its sum to exceed 60.

23


24/25

Answer : 4=n

10. Find the least number of terms of the geometric progression 4,12,36,which

must be taken for its sum to exceed 1 800.Answer : 7=n

11. The sum of the first two terms of a geometric progression is43 and the sum of the

next two terms is16

3, where the common ratio is positive. Find the sum to infinity

of the progression.

Answer : 1=

S

12.

Diagram 1

Diagram 1 shows four circles. Each circle has a radius that is 2 units longer than

that of the previous circle. Given that the sum of the perimeters of these four circles

is 120 cm,

(i) find the radius of the smallest circle.

(ii) the sum of the perimeters from the fifth term to the tenth term.

Answer :(i) r = 12 cm(ii) 300 cm

13. Encik Rahim plans to donate an amount of money to the Rumah Penyayang each

year from 2008. The amount in 2008 will be RM50 000, and thereafter, the amounteach year will be 90% of the amount for the previous year. Calculate

(a) the year in which the donation falls below RM 20 000 for the first time .

(b) the total donation from 2008 to 2015 inclusive

Answer : a) 10=n b) RM284 766.40

24


25/25

14.

Diagram shows two balls in a tube of length 10 m, moving towards each other.Pmoves from one end traveling 60 cm in the first second, 59 cm in the next second

and 58 cm in the third second. Q moves from the other end traveling 40 cm in the

first second, 39 cm in the next second and 38 cm in the third second. The processcontinues in this manner until the two balls meet.

(a) Find the shortest time for the two balls to meet.(give your answer to the nearest

second)

(b) Calculate the distance traveled byP.(c) Calculate the difference in distance traveled by the two balls.

Answer : a) 11s b) 605 cm c) 220 cm

P Q

1. progressions

Documents