The document discusses finding patterns in sequences and determining the nth term of a sequence. It provides examples of sequences with given terms and rules to determine the pattern. The rules include adding a constant, multiplying the term number, squaring the term number, and adding or multiplying the term number. Determining the pattern allows predicting or finding missing terms in a sequence. Examples show writing the next three terms using the identified pattern or rule for the sequence.

1. FINDING THE NTH TERM OF
A SEQUENCE AND
DIFFERENTIATING
EXPRESSION FROM
EQUATION

2. DIRECTION: FIND THE MISSING TERMS IN THE
FOLLOWING PATTERNS. WRITE THE ANSWERS ON
A SHEET OF PAPER.
• A. 3, 6, 9, 12, ____, ____, ____
• B. 0.6, 0.12, 0.18, 0.24, ____, ____,
____
• C. 7, 14, 21, 28, ____, ____, ____
• D. 10, 20, 30, 40, ____, ____, ____
• E. 9, 14, 19, 24, ____, ____, ____
• F. 5, 8, 11, 14, ____, ____, ____
• G. -6, -4, -2, 0, ____, ____, ____
• H. 15, 25, 35, 45, ____, ____, ____
• I. 1, 6, 11, 16, ____, ____, ____
• J. 12, 25, 37, 50, ____, ____, ____

3. NUMBERS, FIGURES, OBJECTS, OR SYMBOLS
ARRANGED IN A DEFINITE ORDER OR SEQUENCE
IS OFTEN ENCOUNTERED IN MATHEMATICS.
• For instance,
• (a) 2, 4, 6, … (b) (c) A, D, G, J, …
• Numbers in (a), figures in (b), and letters in (c) are arranged in a definite
order. Using the given sequences:
• 1. What is the next number in (a)?
• 2. What is the next figure in (b)?
• 3. What is the next letter in (c)?

4. THE NTH TERM OF A SEQUENCE
•A sequence is a set of numbers written in a
special order by the application of a definite
rule. Each number in the sequence is called a
term. Patterns and rules will help us to continue
a given sequence of numbers, figures, or to fill
in the missing numbers or symbols.

5. EXAMPLE 1
Sequence Rule
Nth term
rule
Next three
terms
A. 3, 6, 9, 12, …
Every term after the first is obtained
by adding 3 to the number preceding
it.
(0 + 3), (3 + 3), (6 + 3), (9 + 3), …
or
Multiples of three.
(3 x 1), (3 x 2), (3 x 3), (3 x 4), …
3n 15, 18, 21

6. B. 1, 4, 9, 16, …
Multiply the counting numbers by itself or
squaring counting numbers.
(1 x 1), (2 x 2), (3 x 3), …
n2 25, 36, 49
C.
1
2
,
1
3
,
1
4
, …
The numerator in this sequence is
constant, the denominator is obtained by
adding1 after the other.
1
1+1
,
1
1+2
,
1
1+3
, …
1
𝑛 + 1
1
5
,
1
6
,
1
7
D. 3, 1, -1, -3, …
Every term after the first is obtained by
adding (-2) to the number preceding it.
3, [3 + (-2)], [1 + (-2)], [-1 + (-2)], …
-2(n – 1) + 3
or 5 – 2n
-5, -7, -9

7. EXAMPLE 2
Look for a pattern then write the next three terms.
Sequence Next three terms
•a, 2b, 3c, 4d 5e, 6f, 7g
•a + b, a + 2b a + 3b, a + 4b, a + 5b
•7a, 5a, 3a a, -a, -3a

8. Example 3
Use counting numbers to easily find the rule.
Numbers Rule
A. 2, 3, 4, 5 n + 1
1 + (1) = 2
2 + (1) = 3
3 + (1) = 4
4 + (1) = 5

9. B. 2, 4, 6, 8 2 x n or 2n
2 x (1) = 2
2 x (2) = 4
2 x (3) = 6
2 x (4) = 8

10. C. 2, 5, 8 ,11 3 x n -1
3 x 1 – 1 = 2
3 x 2 – 1 = 5
3 x 3 - 1 = 8
3 x 4 – 1 = 11