The document outlines objectives for a Grade 10 mathematics lesson on sequences, focusing on defining the terms and finding mathematical expressions. It provides activities to identify terms in various sequences and derives rules for generating them. The content includes examples of finite and infinite sequences, along with exercises for determining specific terms based on general formulas.

1. Grade 10 – Mathematics
Quarter I
PATTERNS IN SEQUENCES

2. OBJECTIVES:
• define sequence, finite and infinite
sequence;
• list the next few terms given several
consecutive terms of a sequence; and
• derive a mathematical expression
(rule) for generating the sequence by
pattern searching.

3. ACTIVITY
1. A, D, G, J, ____, ____, ____
2. 1, 3, 5, 7, ____, ____, ___
3. 1, 4, 9, 16, 25, ____, ____, ____
4. 5, 15, 25, 35, ____, ____, ____
5. 1, 3, 6, 10, ____, ____, ____
M P S
9 1311
36 49 64
45 55 65
15 21 28

4. A sequence is a function whose
domain is the set of positive
integers. It also means an ordered
list of numbers.

5. A sequence is infinite if its domain is the
set of positive integers without a last term, {1,
2, 3, 4, 5, ...}. The three dots show that the
sequence goes on and on indefinitely.
A sequence is finite if its domain is the set
of positive integers {1, 2, 3, 4, 5, …, n} which
has a last term, n.

6. Each number in a sequence is
called a term.
Example:
5, 15, 25, 35, 45.
1st
2nd
3rd
4th
5th
5 termsn

7. In the sequence 1, 3, 6, 10, 15,… we can
denote the terms as follows:
𝑎1 = 1 𝑎2 = 3 𝑎5 = 15𝑎4 = 10𝑎3 = 6
Rule: 𝒂 𝒏 =
𝒏
𝟐
(𝒏 + 𝟏)
𝑎1 =
1
2
(1 + 1) = 1
𝑎2 =
2
2
2 + 1 = 3
𝑎3 =
3
2
(3 + 1) = 6
𝑎4 =
4
2
4 + 1 = 10
𝑎5 =
5
2
(5 + 1) = 15
2 4
53
6

8. Find the first 5 terms of the sequence whose general term is
given by 𝒂 𝒏 = 𝒏 − 𝟑 𝒏
.
_____, _____, _____, _____, _____
1st 2nd
3rd 4th 5th
𝑎 𝑛 = 𝑛 − 3 𝑛
𝑎1 = 1 − 3 1
= −2 1
= - 2
𝑎2 = 𝑛 − 3 𝑛
𝑎2 = 2 − 3 2
= −1 2
= 1
−2 1

9. Find the first 5 terms of the sequence whose general term is
given by 𝒂 𝒏 = 𝒏 − 𝟑 𝒏
.
_____, _____, _____, _____, _____
1st 2nd
3rd 4th 5th
−2 1
𝑎 𝑛 = 𝑛 − 3 𝑛
𝑎3 = 3 − 3 3
= 0 3
= 0
𝑎 𝑛 = 𝑛 − 3 𝑛
𝑎4 = 4 − 3 4
= −1 4
= 1
0 1

10. Find the first 5 terms of the sequence whose general term is
given by 𝒂 𝒏 = 𝒏 − 𝟑 𝒏
.
_____, _____, _____, _____, _____
1st 2nd
3rd 4th 5th
−2 1 0 1
𝑎5 = 𝑛 − 3 𝑛
𝑎5 = 5 − 3 5
= 2 5
= 32
32

11. Find the first 4 terms and the 20th term of the sequence
whose general term is given by 𝒂 𝒏 =
−𝟏 𝒏
𝟐𝒏−𝟏
.
𝑎2 =
−1 2
2(2)−1
=
1
3
_____, _____, _____, _____, …, _____
1st 2nd
3rd 4th
20th
𝑎1 =
−1 1
2(1) − 1
=
−1
1
= −1
−1
1
3

12. Find the first 4 terms and the 20th term of the sequence
whose general term is given by 𝒂 𝒏 =
−𝟏 𝒏
𝟐𝒏−𝟏
.
𝑎4 =
−1 4
2(4) − 1
=
1
7
𝑎3 =
−1 3
2(3) − 1
=
−1
5
_____, _____, _____, _____, …, _____
1st 2nd
3rd 4th
20th
−1
1
3
−
1
5
1
7

13. Find the first 4 terms and the 20th term of the sequence
whose general term is given by 𝒂 𝒏 =
−𝟏 𝒏
𝟐𝒏−𝟏
.
_____, _____, _____, _____, …, _____
1st 2nd
3rd 4th
20th
−1
1
3
−
1
5
1
7
1
39
𝑎20 =
−1 20
2(20) − 1
=
1
39
−𝟏 𝒏
the general term causes
the signs of the terms
to alternate between
positive and negative

14. For each sequence, make a guess at the
general term.
1, 8, 27, 64, 125, … 𝑎 𝑛 = 𝑛3
1,
1
2
,
1
3
,
1
4
,
1
5
, … 𝑎 𝑛 =
1
𝑛
-5, 10, -15, 20, -25, … 𝑎 𝑛 = (−1) 𝑛
5𝑛

15. For each sequence, make a guess at the
general term.
1, 4, 9, 25, … 𝑎 𝑛 = 𝑛2
3, -6, 9, -12, 15, … 𝑎 𝑛 = (−1) 𝑛 − 3𝑛
-2, 4, -8, 16, … 𝑎 𝑛 = (−1) 𝑛
2 𝑛