Page 21 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series
6. Find the geometric mean of each of the following numbers.

(a) 3 and 27 (b) 1 and 1 (c) 10 and 10 27
3
3 27
7. Given that the geometric mean of 4p – 3 and 9p + 4 is 6p – 1, find the values of p.
8. The second and fifth terms of a geometric series are 405 and –120 respectively. Find the seventh term
and the sum of the first seven terms of the series.
9. In a geometric progression, the second term exceeds the first term by 20 and the fourth term exceeds the
second term by 15. Find the possible values of the first term.

1
1
3
2 10. Find the sum of the first n terms of the series 12 + + + … . Find also the number of terms required
4
4
such that the sum exceeds 100.
11. A geometric series has first term 16 and common ratio 3 . If the sum of the first n terms exceeds 60,
find the smallest value of n. 4
Sum of an infinite geometric series

When the number of terms of a geometric series is infinite, it is called an infinite geometric series.
Consider the infinite geometric series
1
1 + 1 + 1 + … + 1 2 n – 1 + …
2 4 2
The sum of the first n terms is
1
n
3
1 1 – 1 2 4
S = 2
n
1 – 1
2
1
n
3
= 2 1 – 1 2 4
2
1
As n increases, 1 2 n → 0 and S → 2.
n
2
lim
S = 2.
We say the limit of S as n → ∞ exists and equals 2, and we write n → ∞ n
n
This series is called a convergent series with a sum of 2. Notice that the common ratio is 1 , 1.
2
We now consider the infinite geometric series
1 + 2 + 4 + … + 2 + …
n
The sum of this series is infinite.
This series is called a divergent series.
In general, for a geometric series with first term a and common ratio r,
n
S = a(1 – r )
n
1 – r
= a – 1 a 2 r n
1 – r 1 – r

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02 STPM Math T T1.indd 108 3/28/18 4:21 PM