Download - 1. Progressions
-
7/31/2019 1. Progressions
1/25
1
SM SAINS SERI PUTERI, KUALA LUMPUR
PROGRESSIONS
NAME:
CLASS: .
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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:
1. Skill Practice 4 (page 5)
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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. Skill Practice 5 (page 7)
2. (a) Find the sum of the following arithmetic progression 4, 8, 12, , 60.
7
-
7/31/2019 1. Progressions
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]
2. SPM 2004: Paper 1
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
-
7/31/2019 1. Progressions
9/25
3. SPM 2004: Paper 1
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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
-
7/31/2019 1. Progressions
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