Page 107 - Mathematics_Form_3

Sequences and series

2
11. The first term of a geometric Therefore, S = G + G r + G r + ···
n 1 1 1
n−1
progression is 5, and the common + G r (1)
1
th
ratio is 3. Find the 8 term. Multiply both sides of equation (1) by
12. The third term of a geometric r to get,
FOR ONLINE READING ONLY
progression is 16 and the sixth
3
r S = G r + G r + G r + ··· + G r (2)
2
n
1
1
1
n
1
term is 128. Find the common
ratio and the first term.
Subtract equation (2) from equation (1)
13. The first term of a geometric to obtain,
progression is x, and the common
S − rS = G − G r n (3)
th
ratio is y. Express the n term in n n 1 1
terms of x, y, and n. Factorization of S and G from equation
n
1
14. A geometric progression has a first (3) gives,
term of 2 and a common ratio of S (1 − r ) = G (1 − r ) .
n
1
n
1
3 . Find the smallest value of n for Make S the subject to obtain,
which the n term is less than 0.01. n
th
G (1 − r )
n
1 _
15. If the sum of the first three terms S = 1 − r (4)
n
of a geometric progression is 21
| |
where r is a fraction such that r < 1.
and their product is 216, find the
possible values of the first term Also, subtracting equation (1) from
and the common ratio. equation (2) gives,
16. Find the general term of a GP r S − S = G r − G (5)
n
n n 1 1 n 1
whose seventh term is 13 and tenth
Mathematics for Secondary Schools Sum of the first n terms of a geometric Divide both sides by r − 1 to obtain, (6)
Factorize equation (5) to get,
term is 104.
S (r − 1 ) = G ( r − 1)
n

n
G ( r − 1)
1 _
S =
for r > 1

progression
| |
r − 1
n
Let the sum of the first n terms of a
Therefore, given a geometric progression
geometric progression be denoted by
S . That is, S = G + G + G + ··· + G .
1
1
n
ratio r, the sum of the first n terms is
Since it is a geometric progression, then
G (1 − r )
n
G = G ,G = n Gr ,G = 2 Gr 2 3 , , n with the first term G and the common
, for r < 1
_
1

given by S =
1
1
3
1
2
1
n 1 − r | |
G = Gr n− 1 . G ( r − 1)
n
n
1
1

or S = _ , for r > 1.
| |
n r − 1
100 Student\s Book Form Three
18/09/2025 09:59:25
MATHEMATIC F3 SB.indd 100 18/09/2025 09:59:25
MATHEMATIC F3 SB.indd 100