Page 13 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Example 9

th
The n term of an arithmetic progression is 1 (5n – 3). Obtain the first two terms and also the common
difference of the arithmetic progression. 10
1
Solution: The n term, u = 10 (5n – 3)
th
n
1
The 1 term, u = 10 (5 – 3)
st
1
= 1
5
2 The 2 term, u = 10 (5 × 2 – 3)
1
nd
2
= 7
10
The common difference, d = u – u 1
2
= 7 – 1
10 5
= 1
2

Sum of a finite arithmetic series

When the terms of an arithmetic progression are added up, we will obtain an arithmetic series.
Consider the following arithmetic series made up of 10 terms, i.e.
S 10 = 3 + 6 + 9 + 12 + … + 27 + 30 ………… 
By rewriting the terms of the series backwards, we obtain
S = 30 + 27 + 24 + 21 + … + 6 + 3 ………… 
10
Adding,  + : 2S = 33 + 33 + 33 + 33 + … + 33 + 33
10
= 33 × 10 (since the series has 10 terms)
= 330
∴ S = 165
10
th
For any arithmetic series with first term a, common difference d and n term l, the series can be written as
S = a + (a + d) + (a + 2d) + … + (l – d) + l ………… 
n
By rewriting the terms of the series backwards, we obtain
S = l + (l – d) + (l – 2d) + … + (a + d) + a ………… 
n
 + : 2S = (a + l) + (a + l) + (a + l) + … + (a + l) + (a + l)
n
= n(a + l) since the series has n terms
n
∴ S = — (a + l).

n 2
Since the n term, l = a + (n – 1) d,
th
S = — [a + a + (n – 1) d]

n
n 2 Arithmetic

n
i.e. S = — [2a + (n – 1) d] Sequence
n 2 VIDEO
Both these formulae derived can be used to find the sum of the first n terms of an arithmetic series.

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02 STPM Math T T1.indd 100 3/28/18 4:21 PM