This document discusses arithmetic and geometric sequences. An arithmetic sequence is one where the difference between consecutive terms is constant, while a geometric sequence is one where the ratio between consecutive terms is constant. Formulas are provided to calculate individual terms and the sum of terms for both arithmetic and geometric sequences. Examples are worked through demonstrating how to identify sequences and apply the formulas to problems involving cash prizes, savings accounts, and other real-world scenarios.

1. Syed Ashraaf Bin Wan Mohamad
International College of Advance Technology Sarawak (iCATS)

2. Syed Ashraaf Bin Wan Mohamad
International College of Advance Technology Sarawak (iCATS)
At the end of this Chapter, you will be able to;
1. Explain the terms arithmetic & geometric sequences ?
2. Identify arithmetic & geometric sequences ?
3. Calculate the terms in arithmetic & geometric sequences ?
4. Calculate the sum of terms in arithmetic & geometric sequences ?
5. Apply the concepts of arithmetic & geometric sequences to some common in daily life ?

3. 1.1 Introduction
 A sequence of a progression is a succession of terms,
T1, T2, T3, … Tn , ( exp. 5, 6, 7, 8… )
 Thus a sequence is a list of numbers arranged in a specified order.
 A series is the indicated sum,
T1 + T2 + T3 + … , ( exp. 5 + 6 + 7 + 8… )
There are many types of sequences but this chapter will only discuss
two sequences : -
1. ARITHMETIC SEQUENCES &
2. GEOMETRIC SEQUENCES.

4. 1.2 Arithmetic Sequence
 Arithmetic sequence is one in which the difference between any term
and the preceding term is the same throughout.
 This common difference (d), can be obtained by subtracting any term
from the term which immediately follows it.
 Thus, d = T2 – T1 = T3 – T2 = T4 – T3, etc.
 Examples:
Arithmetic sequence Common difference, d
a) 4, 8, 12, 16, ….. 4
b) 7, 5, 3, 1, ……. -2
c) -5, -9, -13, -17, ……. -4
d) 1
2
, 1, 1
1
2
, 2
1
2

5. What if…??
Find the value of and .
Find the value of and .
Can you find ?

6.  If the first term of an arithmetic sequence is a, and the common
difference is d, then the arithmetic sequence can be written as
a , a + d , a + 2d , a + 3d
 To find the nth term:
𝑇𝑛 = 𝑎 + 𝑛 − 1 𝑑
Where:
𝑇𝑛 = nth term,
𝑎 = first term,
𝑛 = number of terms, and
𝑑 = common difference.
 To find the sum of the first nth term:
𝑆𝑛 =
𝑛
2
[ 2𝑎 + 𝑛 − 1 𝑑 ]

7.  Example 1;
Giving the following arithmetic sequence: 2, 10, 18, ….., find
i) The tenth term.
ii) The sum of the first ten terms.
Answer:
i) 𝑇10= 2 + 10 − 1 8
= 74
ii) 𝑆10 =
10
2
[ 2(2) + 10 − 1 8 ]
= 380

8.  Example 2:
In a contest, all ten finalist were given cash prizes. The first winner
was given RM800, the second RM740, the third RM680 and so on.
Calculate the total amount of money awarded to all the finalists.
Answer:
𝑆10 =
10
2
[ 2(𝑅𝑀800) + 10 − 1 − 60 ]
= RM 5300.

9. 1.3 Geometric Sequence
 A geometric sequence is a sequence in which the ratio of each
term to the preceding term is the same throughout.
 Examples:
Geometric Sequence Common ratio, r
a) 1, 2, 4, 8, …… 2
b) 3, 9, 27, 81, ….. 3
c) -5, 10, -20, 40, ….. -2
d) 20, 10, 5,
5
2
, … . . 1
2

10.  If the first term of a geometric sequence is a and the common ratio is r,
then the geometric sequence can be written as 𝑎, 𝑎𝑟, 𝑎𝑟2, 𝑎𝑟3, … . .
 To find the nth term:
𝑇𝑛 = 𝑎𝑟 𝑛−1
Where:
𝑇𝑛 = nth term,
𝑎 = first term,
𝑟 = common ratio, and
n = number of terms.
 To find the sum of the first nth term:
𝑆 𝑛 =
𝑎 (𝑟 𝑛 − 1)
𝑟 − 1
𝑖𝑓 𝑟 > 1 𝑆 𝑛 =
𝑎 (1 − 𝑟 𝑛)
1 − 𝑟
𝑖𝑓 𝑟 < 1

11.  Example 1:
Given the following geometric sequence: 5, 15, 45, 135, …, find
i) The eight terms.
ii) The sum of the first eight terms.
Answer:
i) 𝑇8 = 5 𝑥 38−1
= 10935
ii) 𝑆8 =
5 (38−1)
3 −1
= 16400

12.  Example 2:
Maimunah saves RM1000 in a savings account that pays 8%
compounded annually. Find the amount in her account at the 5 years.
Answer:
Note that :– Year 0 = 1000
Year 1 = 1000 (1.08)
Year 2 = 1000 (1.08)² and so on.
a = 1000; r = 1.08; n = 5
𝑇5 = 1000 𝑥 1.085−1
= RM1360.49

13. 1. Can you explain the terms arithmetic & geometric sequences ?
2. Can you identify arithmetic & geometric sequences ?
3. Are you able to calculate the terms in arithmetic & geometric
sequences ?
4. Are you able to calculate the sum of terms in arithmetic & geometric
sequences ?
5. Can you apply the concepts of arithmetic & geometric sequences to
some common in daily life ?
THE END

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