Page 24 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Substituting the values a = 2 and r = 0.95.
2(0.95) n , 1
1 – 0.95
n
(0.95) , 0.025
n log (0.95) , log (0.025)
10
10

n(–0.0223) , –1.6021
n  –1.6021 = 71.8
– 0.0223
i.e. the smallest value of n such that S – S , 1 is 72.
n
∞
2
Exercise 2.4

1. Determine whether each of the following series is convergent or otherwise.
(a) 2 + 2 + 2 + … (b) 18 + 15 + 12 + … (c) 40 – 20 + 10 – …
3 3 2
1
3
(d) 4 + 8 + 16 + … (e) k + 2k + 3k + … (f) 7 – 3 + 1 – …
5 15 45 2 4
2. Find the sum of each of the following geometric series.
(a) 6 + 2 + 2 + … (b) 1 – 1 + 1 – …
3 2 4
(c) 10 + 1 + 0.1 + … (d) 45 – 30 + 20 – …

3. Express each of the following recurring decimals as an infinite geometric series or as the sum of a constant
and an infinite geometric series. Hence, express each of the decimals as a fraction in its simplest form.
· · · · · · · ·

(a) 0.48 (b) 0.072 (c) 0.5813
· · · · · ·

(d) 0.3354 (e) 0.9218
4. Find the sum of each of the following infinite geometric series and state the range of values of x for which
the result is valid.
2
(a) 1 + 2x + 4x + … (b) 1 – 1 x + 1 x – …
2
2 4
2
3
(c) 3 – 6x + 12x – … (d) x + 1 x + 1 x + …
2
3 9
5. The sum of the first n terms of a geometric series is 2 n (3 – 1). Obtain the first three terms and the sum
n
to infinity of the series. 3
6. Find the value of r such that the sum of the series
2
1 + r + r + … + r n – 1 + …
is twice the sum of the series
2
3
1 – r + r – r + …
7. A ball rebounds to a height five-eighths of its previous height above the ground. If a ball is dropped from
a height of 3 metres, find the total distance travelled by the ball before it comes to rest.
1
8. The first three terms of a geometric series are 2, – and 1 respectively. Find the sum to infinity of the
2 8
series. Find the smallest value of n such that the difference between the sum of the first n terms and the
sum to infinity is less than 10 .
–5

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02 STPM Math T T1.indd 111 3/28/18 4:21 PM