Unit3 progressions[1]

BA301 : ENGINEERING MATHEMATHICS 3
PROGRESSIONS

3.1 INTRODUCTION

 A sequence is a set of terms which are written in a definite order obeying a certain rule.
Example of a sequence is 2,4,6,8,.... which is a sequence of even numbers.
 A series is the sum of the terms of a sequence. Example of a series is 1+ 3 + 5 + 7...

3.2 ARITHMETIC PROGRESSION (AP)

3.2.1 THE nth TERM OF AN AP

 An AP is a sequence of numbers where each new term after the first term,a is made by
adding on a constant amount to the previous term.
 This constant is known as the common difference can be positive, zero or negative

The common difference,d is given by : The nth term, Tn of an AP is given by :

d = Tn + 1 - Tn Tn = a + (n -
1) d
Where: Tn + 1 = (n+1) th term Where: a =
first term
T n = n th term d=
common difference

 An AP can be written as a, a+d, a+2d, a+3d,....

Example 1:
a) Calculate the 8th term of an AP: 3, 7, 11, 15, 19, ....

7
b) Calculate the 5th term of an AP: 3, , 4, ....
2

Anisah Ahmad JMSK Jun 2011


c) Find the 7th and 18th terms of the AP 1.3, 1.1, 0.9, ….

Example 2:
1 1 3
a) What is the number of terms in the AP 1 ,2 ,3 ,.....,10 .
4 2 4

b) The 5th term and 6th term of an AP are 27 and 21 respectively. Find
the first term.

c) Find the nth term of the AP 14, 9, 4, ….. in terms of n. Hence find the
21st term of the AP.

d) The 3rd and 5th terms of an AP are -10 and -20 respectively. Find the:
i) First term ii) The 12ve term



Exercise 1:
1) Find the 6th and the 15th terms of the AP:
a. 1, 5, 9, ……
b. -4, -7, -10, ……
1 1 2
c. , , ,.....
3 2 3
2) Find the number of terms in the following AP:
a. 3, 10, 17, …… , 143
b. -143, -130, …… , 221
5 7 9 33
c. , , ,....,
4 4 4 4

3) Given that the 4th term of an AP is 12 and the common difference is
2. Find the first 3 terms of the AP.

4) Find the number of terms of the AP 407, 401, 395, ….. , -133. Hence,
find the 20th term of the AP.

5) Find the nth term of the AP 21, 15, 9, ….. in terms of n. Hence find the
17th term of the AP.

6) The nth term of an AP is given by Tn = 4 – 9n. Find the first term, 10th
term and common difference.

7) The 3rd and 12ve terms of an AP are -3 and 24 respectively. Find
the:
i) First term and common difference
ii) The 15th term

3.2.2 SUM OF THE FIRST n TERMS OF AN AP

 The sum of the first n terms, Sn of an AP is given by:

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


Where: a = first term
d = common difference

Example 3:
a) Find the sum of the first 20 terms of the AP – 8, – 3, 2, …. .

b) Find the sum of the AP 9.8, 9.2, 8.6, 8.0, …., – 5.2.

c) The sequence 20, 26, 32, … is an AP. Find the sum of the 6th term to
the 15th term of the AP.

d) The sequence 25, 22, 19, …. is an AP. Find the value of n for which
the sum of the first n terms of the AP is 116.



e) The first three terms of an AP are 2p – 1, 4p and 5p + 4. Find:
i) The value of p

ii) The sum of the first 13 terms of the AP

f) The 7th term of an AP is 11 and the sum of the first 16 terms of the
AP is 188. Find:
i) The first term and the common difference of the AP.

ii) The sum of the first 10 terms after the 16th term.



Exercise 2:
1) Find the sum of :
a. The first 9 terms of the AP 7, 11, 15, 19, …. .
b. The first 12 terms of the AP 10, 4, -2, …. .

2) Find the sum of the following AP:
a. 1 + 3 + 5 + …… + 101
b. 205 + 200 + 195 + 190 + ….. + 105

3) The sequence -30, -22, -14, …. is an AP. Find the sum from the 9th
term to the 14th term of the AP.

4) The sequence -9, -5, -1, …. is an AP. Find the value of n for which the
sum of the first n terms of the AP is 90.

5) The first three terms of an AP are k – 3, k + 3 and 2k + 2. Find:
i) The value of k
ii) The sum of the first 9 terms of the AP

6) The 5th and 9th terms of an AP are 37 and 65 respectively. Find:
ii) The sum of the first 15th terms.

7) Given that the sum of the first 4 terms is 34 and the sum of the next 4
terms is 82. Find:

8) The 5th term of an AP is 10 and the sum of the first 10 terms of the
AP is 115. Find:
iii) The sum of the first 10th terms after the 10 terms.



3.2.3 ARITHMETIC MEAN

 If a, b, c are three consecutive terms of an AP, then :

b–a=c–b
b is the arithmetic 2b = a + c
mean of a and c b=a+c
2

Example 4:
1
a) Determine whether the following sequences is an AP: 8, 12 , 17,
2
….

b) 2m, 4m + 1, 14 are the three consecutive terms of an AP. Find the
value of m.

c) Find arithmetic mean for 3 and 25.

d) Find 3 arithmetic means for 3 and 11.



e) Find 4 arithmetic means for 3 and 23.

3.2.4 SOLVING PROBLEMS RELATED TO AP

Example 5:
a) Ismail has RM 1760 in his bank account in January. Starting from
February, he withdraws RM 135 monthly from his bank account. Find
his bank balance at the end of December in the same year.

b) Mr Siva rents a house with a monthly rental of RM420. in the
agreement, it is stated that the monthly rental will increase by the
same amount each year. In the 8th year, Mr Siva has to pay a monthly
rental of RM560. Calculate :
i. The increase in the yearly rental
ii. The total amount of house rental Mr Siva has to pay from the 6th
year to the 12th year.



3.3 GEOMETRIC PROGRESSION (GP)

3.3.1 THE nth TERM OF A GP

 A GP is a number sequence where each term after the first term
is obtained by multiplying the preceding term by a constant known
as the common ratio.
 The common ratio cannot take the values 0 or 1 but can be
positive or negative.

The common ratio,r is given by : The nth term, Tn of
an GP is given by :

r = Tn+1 Tn = ar n – 1
Tn Where: a =
first term
Where: T n+1 = (n+1) th term r=
common ratio
Tn = n th term

 A GP can be written as a, ar, ar 2, ar 3, .....

Example 6:
a) Calculate the 7th and 15th terms of the GP: 4, 12, 36, ....



3 1 1
b) Calculate the 6th and 10th terms of the GP: , , ,....
2 2 6

1
c) What is the number of terms in the GP 81, 27, 9, …., ?
27

1 1 1
d) Given that the nth term of the GP , , , ….. is 128. Find the
32 8 2
value of n.

e) Find the nth term of the GP -8, -4, -2, …. in terms of n. Hence, find the
6th term of the GP.



f) The 2nd and 5th terms of a GP are 1 and -8 respectively. Find :
i) The first term and the common ratio

ii) The 7th term

Exercise 3:
1) Find the nth and 8th term of the GP :
a. -3, 9, -27, 81, ......
b. 16, -8, 4, 2, ……..

2) For the following GP, calculate the value for the 10th and 15th terms :
a. 1,3,5,7,.........
b. 2,4,8,16,........

1
3) What is the number of terms in the GP 256, 64, 16, …., ?
64

243
4) Given that the nth term of the GP 64, 48, 36 ….. is . Find the
16
value of n.

9 3
5) Find the nth term of the GP , , 1, …. in terms of n. Hence, find the
4 2
8th term of the GP.

6) Given that x + 10, x + 1 and x are the first 3 terms of a GP. Find the
value of x and the common ratio.


7) For a GP, the 3rd term exceeds the first term by 9 and the sum of the
2nd and 3rd terms is 18. Find :
i) The first term and the common ratio.
ii) The 6th term

3.3.2 SUM OF THE FIRST n TERMS OF A GP

 The sum of the first n terms, Sn of a GP is given by :

Sn = a ( 1 – r n )
1–r
Where: a = first term
r = common ratio


Example 7:
a) Calculate the sum of the first 7 terms of the GP 4, 10, 25, ....

1
b) Calculate the sum of the first 5 terms of the GP 5, 1, , ....
5

3 3 3
c) Calculate the sum of the GP , , , ...., 6.
8 4 2

d) The sum of the first n terms of the GP is 189. Given that the first term
and the 2nd term are 3 and 6 respectively. Find the value of n.

e) Given a GP 27, 9, 3,…., find the sum from the 4th term to the 9th term
of the GP.



f) The first three terms of a GP are k + 1, k - 3 and k - 6. Find :
i) The value of k

ii) The sum of the first 6 terms of the GP

Exercise 4:
1) Find the sum of the first 6 terms of the GP :
a. -2, -1, -1/2
b. 2, -6, 18

2) The sum of the first n terms of the GP 27, 21, 15, …. is – 72. Find the
value of n.

25
3) The 4th and 7th terms of a GP are 25 and respectively. Find the
8
sum of the first 3 terms of the GP.
4) Given a GP 10, 8, 6.4,…., find the sum from the 3rd term to the 7th
term of the GP.

1031
5) The sum of the first 5 terms of a GP is and the common ratio is
125

2
. Find :
5
i) The first term
ii) The sum of the first 5 terms
201
6) For a GP, the 2nd and 5th terms are -6 and . Find :
4
i) The common ratio

ii) The sum of the first 5 terms

7) The first three terms of a GP are h + 2, h - 4 and h - 7. Find :
i) The value of h
ii) The sum of the first 8 terms of the GP
3.3.3 SUM TO INFINITY OF A GP

 For the case of -1 < r < 1, the sum of the first n terms of
a GP is given by :

S∞ = a
S∞ is read as
1–r
sum to infinity

Example 8:
a) Find the sum to infinity of each of the following GP :
5 5
a. 5, , , ....
3 9

1 4 1 1
b. 1, , , , , ....
2 3 4 8

1 1
c. 4, -2, 1, − , ,
2 4



b) The sum to infinity of a GP with a first term of 12 is 9.6. Find the
common ratio of the GP.

3
c) The sum to infinity of a GP with a common ratio of is 32. Find the
4
first term of the GP.

16
d) For a GP, the first term is 18 and the 4th term is . Find :
3
i. The common ratio

ii. The sum to infinity of the GP



e) For a GP, the first term exceeds the 2nd term by 8 and the sum to
infinity is 18. Find the common ratio of the GP

3.3.4 GEOMETRIC MEAN

 If a, b and c are three consecutive terms of a geometric
progression then :

b=c
a=b
b is the b 2 = ac
geometric mean
b = √ ac
of a and c

Example 9:
a) Determine whether the following sequences is a GP: 3, 9, 27, 81, ….

b) Given that m - 8, m - 4, m + 8 are the three consecutive terms of a

GP. Find the value of m.

c) Find 3 geometric mean for 3 and 23

d) Find 3 geometric mean for 10 and 100 000

e) Find 5 geometric mean for 4 and 2916



3.3.4 SOLVING PROBLEMS RELATED TO GP

Example 10:
a) A particle moves along a straight line from point P. it covers a
distance of 64 cm in the 1st second, 48 cm in the 2nd second, 36 cm
in the 3rd second and so on. Find its distance from point P 9 seconds
later.

b) The price of a house in a certain residential area is RM 220 000. Its
price increases 5% each year. Calculate the minimum number of
years needed for the price to be more than RM 400 000 for the
first time.

c) Ibrahim saves 5 cent on the first day, 10 cent on the second day, 20
cent on the third day and so on such that the amount of money he
saves on each day is twice that of the previous day. Calculate the

minimum number of days needed for the total amount of money to be
more than RM 1000 for the first time.


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