Page 23 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Example 19

Express the recurring decimal 0.327 as an infinite geometric series. Hence, express 0.327 as a fraction in
its simplest form.
(Note: The recurring decimal 0.327 = 0.327327327…)

Solution: 0.327 = 0.327327327…
= 0.327 + 0.000327 + 0.000000327 + …
= 0.327 + 0.327 × 10 + 0.327 × 10 + …
–3
–6
–3
This is an infinite geometric series with a = 0.327 and r = 10 = 0.001.
2 Hence, 0.327 = 0.327 a
1 – 0.001 Using S = –––––
∞
1 – r

= 0.327
0.999
= 327
999
= 109
333

Example 20

Find the sum of the series 1 + 3x + 9x + 27x + … , stating the range of values of x for which the result
2
3
is valid.
2
3
Solution: 1 + 3x + 9x + 27x + …
is a geometric series with a = 1 and r = 3x.
1
Hence, the sum to infinity is S = 1 – 3x .
∞
The result is valid if |3x| , 1,
i.e. |x| , 1
3
1
or – , x , 1
3 3

Example 21

The first term of a geometric series is 2 and the common ratio is 0.95. The sum of the first n terms of this
series is S and the sum of this series is S . Find the smallest value of n such that S – S , 1.
n
n
∞
∞
Solution: The first term, a = 2
Common ratio, r = 0.95 n
a
Using the formulas S = a(1 – r ) and S = 1 – r ,
∞
1 – r
n
S – S , 1
∞
n
n
⇒ a – a(1 – r ) , 1
1 – r 1 – r
a – a + ar n , 1
1 – r 1 – r 1 – r
ar n , 1
1 – r
110

02 STPM Math T T1.indd 110 3/28/18 4:21 PM