Page 10 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Example 5

Express each of the following series by using the summation notation ∑.
(a) 1 + 4 + 9 + 16 + … + 100
(b) 1 · 2 + 2 · 3 + 3 · 4 + … + 19 · 20

2
2
2
2
2
Solution: (a) 1 + 4 + 9 + 16 + … + 100 = 1 + 2 + 3 + 4 + … + 10
th
∴ the r term, u = r 2 10
r
Hence, the series may be written as ∑ r .
2
r = 1
(b) The r term, u = r(r + 1). 19
th
r
Hence, the series may be written as ∑ r(r + 1). 2
r = 1
Example 6
Rewrite each of the following series by using the summation notation ∑.
(a) 2 + 5 + 10 + 17 + … + 401 (b) 1 + 1 + 1 + … + 1
2 3 4 50
Solution: (a) 2 + 5 + 10 + 17 + … + 401
2
2
2
2
= (1 + 1) + (2 + 1) + (3 + 1) + (4 + 1) + … + (20 + 1)
2
th
2
The r term, u = r + 1, (1  r  20)
r
20
Hence, the series may be written as ∑ (r + 1).
2
r = 1
(b) 1 + 1 + 1 + … + 1
2 3 4 50
= 1 + 1 + 1 + … + 1
1 + 1 2 + 1 3 + 1 49 + 1
1
th
The r term, u = r + 1 , 1  r  49.
r
49
Hence, the series may be written as ∑ 1 .
r = 1 r + 1
Example 7
Write down the first three terms and the last term of the following series.
20 10
(a) ∑ r(r + 2) (b) ∑ (–1) r + 1 r
2
r = 1 r = 1
Solution: (a) The r term, u = r(r + 2)
th
r
Thus u = 1(1 + 2) = 3
1
u = 2(2 + 2) = 8
2
u = 3(3 + 2) = 15
3
The last term, u = 20 (20 + 2) = 440
20
(b) The r term, u = (–1) r + 1 r
2
th
r
2 1
Thus u = (–1) 2 = 2
1
u = (–1) 2 = –4
3 2

2
u = (–1) 2 = 8
4 3
3
The last term, u = (–1) 2 = –1024
11 10
10
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02 STPM Math T T1.indd 97 3/28/18 4:21 PM