Page 96 - Mathematics_Form_3

Sequences and series

since there is a common difference of
4 between the numbers. An arithmetic
series is obtained by adding the
terms of an arithmetic progressions.
For instance, arithmetic series with
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common differences 1, − 2, and 4x , are
respectively:
1. 1 + 2 + 3 + 4 + · · · + 99

2. − 1 − 3 – 5 − 7− · · ·
Figure 4.3: Nested squares
3. x + 5x + 9x + 13x + · · ·
3. Study the pattern carefully to
determine the number of sticks or
th
total number of nails covered by The n term of an arithmetic
each rubber bands used in each progression
square. If n is the number of terms of an arithmetic
4. Study the pattern carefully when progression (AP) with the first term A ,
1
increasing the number of nails or its n term is denoted by A . Consider
th
n
sticks as you increase the number a problem of counting in 2’s starting
of nested squares. from one. It means the first term A is
1
5. From the pattern obtained in task 4, 1 and the common difference d is 2. To
derive a general rule for obtaining find the 23 term, do the following steps:
rd
the number of nails covered by

rubber bands or sticks which may A = 1 and d = 2
1
be used as you increase the number A = 1 + d = 1 + 2
of the nested squares. 2

6. If you have 312 sticks or nails, how A = 1 + d + d = 1 + 2d = 1 + (2 × 2)
3
many squares will the final pattern A = 1 + d + d + d = 1 + 3d =
contain? 4
1 + (3 × 2)
7. Prepare a poster giving a summary

of your work and use it in a class A = 1 + d + d + d + d = 1 + 4d =
5
discussion. 1 + (4 × 2) and so on.
th
From the pattern, the 23 term is
A sequence of numbers in which each A = 1 + (22 × 2 ) = 45. Mathematics for Secondary Schools
term after the first is obtained by adding a Similarly, the n term is obtained
23
th
constant number to the preceding term is
called as an Arithmetic Progression. The by adding n − 1 times the common
constant number which is the difference difference 2, to the first term. Thus, A
n
between any two consecutive terms is is given by:
called the common difference, denoted A = 1 + 2(n − 1 ) = 2n − 1.
n
by d. Therefore, the series is
The number of sticks, in Figure 4.3
are 4, 8, and 12. Thus, they form an AP 1 + 3 + 5 + 7 + · · · + (2n − 1 ).

Student\s Book Form Three 89

18/09/2025 09:59:20
MATHEMATIC F3 SB.indd 89
MATHEMATIC F3 SB.indd 89 18/09/2025 09:59:20