Page 18 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series
Geometric series

Consider the sequence of numbers
1, 2, 4, 8, 16, …
Each term (except the first term) in the sequence is obtained by multiplying the previous term with a fixed
number 2.
Thus, this sequence can also be written as
4
1, 1 × 2, 1 × 2 , 1 × 2 , 1 × 2 , …
2
3
This type of sequence is called a geometric progression and the fixed number is called common ratio.
If a geometric progression has first term 3 and common ratio –2, then the terms are
2
3, 3(–2), 3(–2) , 3(–2) , … 2
3

or 3, –6, 12, –24, …
If the first term of a geometric progression is a and its common ratio is r, then the geometric progression may
be represented as
2
a, ar, ar , …, ar n – 1
th
with its n term,
u = ar n – 1
n

Sum of a finite geometric series
When the terms of a geometric progression are added up, we will obtain a geometric series.
Consider the following geometric series which is made up of 10 terms, with the first term 1 and common ratio
5, i.e.
2
3
S = 1 + 1(5) + 1(5) + 1(5) + … + 1(5) 9
10
3
2
9
S = 1 + 5 + 5 + 5 + … + 5 ………… 
10
10
4
2
3
 × 5: 5S = 5 + 5 + 5 + 5 + … + 5 ………… 
10
10
 – : (5 – 1) S = 5 – 1
10
10
S = 5 – 1
5 – 1
10
= 1 (5 – 1)
10
4
For any geometric series with first term a and common ratio r ≠ 1, the sum of the first n terms, S , can be
n
written as
3
S = a + ar + ar + ar + … + ar n – 1 ……… 
2
n
2
n
 × r: rS = ar + ar + ar + ar + … + ar ……… 
4
3
n
n
 – : (r – 1)S = ar – a
n
n
S = a(r – 1) , for r  1
r – 1
n
Geometric
n
or S = a(1 – r ) , for r , 1 VIDEO Sequence
1 – r
n
105
02 STPM Math T T1.indd 105 3/28/18 4:21 PM