Page 16 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Example 14

The sum of the first n terms of an arithmetic progression is given by S = pn + qn .
2
n
Given that S = 6 and S = 11,
3
5
(a) find the values of p and q,
th
(b) deduce, or otherwise, an expression for the n term and the value of its common difference.
Solution: (a) S = pn + qn 2
n
When n = 3, S = 3p + 9q = 6
3
p + 3q = 2 ………… 
When n = 5, S = 5p + 25q = 11
5
p + 5q = 11 …………  2
5
 – : 2q = 1
5
q = 1
10
Substituting q = 1 into ,
10
p + 3 = 2
10
3
p = 2 – 10
= 17
10
Hence, p = 17 and q = 1 .
10 10
(b) S = pn + qn 2
n
Thus S n – 1 = p(n – 1) + q(n – 1) 2
The n term, u = S – S n – 1
th
n
n
= pn + qn – p(n –1) – q(n – 1) 2
2
= pn + qn – pn + p – q(n – 2n + 1)
2
2
= p + 2nq – q
1
= 17 + 2n 1 2 – 1
10 10 10
= 1 (n + 8)
5
The common difference, d = u – u n – 1
n
= 1 (n + 8) – 1 (n – 1 + 8)
5 5
= 1 (n + 8) – 1 (n + 7)
5 5
= 1
5
Check:
u = 1 (2 + 8) = 10
2
5
5
u = 1 (1 + 8) = 9
1
5
5
d = u – u 1
2
= 10 – 9
5 5
= 1
5
103

02 STPM Math T T1.indd 103 3/28/18 4:21 PM