Page 105 - Mathematics_Form_3

Sequences and series

In general, the n term G of any Example 4.18
th
n
geometric progression whose first term is The first term of a geometric progression
G and the common ratio is r is obtained is 4 and the fourth term is 108. Find the
1
by using the following procedure: common ratio.
Terms Exponent of r
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G = G 0 Solution

1
1
G = G r 1 Given G = 4, G = 108, n = 4 .
1
4
2 1 From G = G r .
n−1
G = G r 2 n 1
2
3 1 Substituting the values into the formula
3
G = G r 3
4 1 gives,
G = G r n–1 108 = 4 r .
n−1
3
n 1
Therefore, the n term is G = G r . Dividing by 4 both sides gives,
n−1
th
n 1
3
3
3
r = 27, which implies that r = 3 .
Example 4.17 Since the exponents are equal, equate
the bases. Thus, r = 3.
Find the 8 term of each of the following
th
geometric series: Therefore, the common ratio is 3.
(a) 2 + 4 + 8 + · · · (b) 12 + 6 + 3 + · · ·
Example 4.19
Solution
4 Find the number of terms in the
(a) Given G = 2, r = 2 = 2 and n = 8 . following geometric series.
1
From G = G r substituting 1 + 2 + 4 + 8 + 16 + · · · + 512
n−1
n 1
the values into the formula gives, Solution
G = 2 (2) From the given series, G = 1, r = 2,
7
8 1
and G = 512 . 1 n−1 n−1
= 2
8
Mathematics for Secondary Schools (b) Given G = 12, r = 12 = 1 , n = 8 2 = 512 × 2
n
From G = G r , it implies that
= 256 .
n
512 = 1 × 2
Therefore, the 8 term is 256.
th
n
6
n
10
2 = 2
2
1
n = 10
n−1
Apply the formula G = G r to
1
n
Therefore, the number of terms is 10.
obtain,
7
1 __
G = 12
( )
8
2
12

128
The value of a machine depreciates by
25% every year. If its value is 75,000,000
3 ___
= ____ Example 4.20
=
32 Tanzanian shillings when it is new,
3 ___
th
Therefore, the 8 term is . what is its value after 21 years?
32
98 Student\s Book Form Three
18/09/2025 09:59:24
MATHEMATIC F3 SB.indd 98 18/09/2025 09:59:24
MATHEMATIC F3 SB.indd 98