Page 19 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Example 15

The 5 term of a geometric series is 18 and the 7 term is 54. Given that the sum of the first 10 terms of
th
th
this series is positive, find the 1 term, the common ratio and the sum of the first 10 terms.
st
Solution: Let a be the 1 term and r the common ratio.
st
4
th
the 5 term, ar = 18 ………… 
6
th
and the 7 term, ar = 54 ………… 
2
 ÷ : r = 54 = 3
18
r = ± 3
2 Substituting r = ± 3 into ,
a(9) = 18
a = 2
n
From the formula S = a(r – 1) ,
r – 1
n
10
when r = 3 , S = 2[( 3) – 1]
10
3 – 1
= 2(243 – 1)
3 – 1
= 484
3 – 1
10
When r = – 3 , S = 2[1 – (– 3) ]
10
1 – (– 3 )
2(1 – 243)
=
1 + 3
= – 484
3 + 1
Since it is given that S  0, thus
10
484
a = 2, r = 3 and S = 3 – 1 .
10
Example 16

The first term of a geometric series is 5 and the common ratio is 1.5. Find the number of terms needed
such that the sum of the series exceeds 200.
Solution: Given that a = 5 and r = 1.5
The sum of the 1 n terms is
st
5[(1.5) – 1]
n
n
S = 1.5 – 1 = 10[(1.5) – 1]
n
For S to exceed 200,
n
n
10[(1.5) – 1]  200
n
(1.5) – 1  20
n
(1.5)  21
n log 1.5  log 21
10
10
log 21
n  10
log 1.5
10
= 7.5
Hence, the smallest value of n is 8, i.e. 8 terms are needed such that its sum exceeds 200.
106

02 STPM Math T T1.indd 106 3/28/18 4:21 PM