The document provides information about sequences, including defining characteristics, types of sequences, general terms of sequences, recursive and explicit formulas, and examples. It discusses finite and infinite sequences, terms in a sequence, writing the general nth term as a function, and finding specific terms. Examples of sequences include the Fibonacci sequence and using the golden ratio in photography. Worked problems demonstrate finding terms of sequences given their formulas.

1. FIRST QUARTER
MATHEMA
TICS
MELC: Generates patterns…

3. In
History,
Philip
pine
Asian World
In
Biology,
Cells Tissue
s
Organs
In Essay
Writing,
Introduc
tion
Body Conclusi
on

4. Learn Mathematics online, the best ways we can

5. 1. find the pattern of a sequence.
2. identify the next term of a sequence.
3. differentiate between infinite and finite sequences.
4. give the value of sum of numbers in a sequence
5. value the importance of respect in a family and orderliness
in workplace.
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At the end of this course presentation, you will be able to:

7. Sequence
A succession of numbers in a
specific order
1, 2, 3, 4, 5,…

9. Where can sequence be
observed in daily living?
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10. Where can sequence be
observed in your own
home?
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11. Where can sequence be
observed in a specific
household chore?
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13. Term
1, 2, 3, 4, 5
numbers in the sequence
1st term 5th / last term

14. Types of Sequences

15. Finite Sequence
a sequence that ends
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
+1 +1 +1 +1 +1 +1 +1 +1 +1
< A sequence that has a first term and
a last term >
limited domain

16. Finite Sequence

17. Infinite Sequence
when the sequence goes on forever
it is called an infinite sequence
{1, 2, 3, 4, 5, …} ellipsis
<There is another term, after each term of the
sequence>
infinite domain

18. Determine…
1. 1, 4, 16, 25, … , 81
2. 8, 16, 24, 32. …
3. 1, 4,7,10,13, …
4. 6, 12, 18, 24, … , 60
5. 1,5, 9, 13, 17, …

20. General terms of
sequences
If a is a sequence, its general term or nth
term is its function value a(n) usually
denoted by an (read as “a sub n”)

21. In the sequence 0, 1, 3, 6, 10, 15,… we
can denote the terms as follows:
a₁ = 0
a₂ = 1
a₃ = 3
a₄ = 6
a₅ = 10
a₆ = 15
1st term 2nd term 3rd term 4th term 5th term
𝑎1 𝑎2 𝑎3 𝑎4 𝑎5
6th term 7th term 8th term 9th term 10th term
𝑎6 𝑎7 𝑎8 𝑎9 𝑎10
nth term
𝑎𝑛

22. Find the first four terms of the
sequence defined by
an = 2n + 1
first term
Replace n with 1, 2, 3, and 4 in the
expression of an
a₁ = 2(1) + 1
= 3
second term
a₂ = 2(2) + 1
= 5
third term
a₃ = 2(3) + 1
= 7
20th term
a₄ = 2(4) + 1
= 9
an = 2n + 1
= 3, 5, 7, 9
fourth term
A₂₀ = 2(20) +
1
= 41

23. an =
𝒏+𝟐
𝒏+𝟒
Find the first five terms
a₁ =
𝟑
𝟓
a₂ =
𝟐
𝟑
a₃ =
𝟓
𝟕
a₄ =
𝟑
𝟒
a₅ =
𝟕
𝟗
a₂₅=
𝟐𝟕
𝟐𝟗
a₃₇=
𝟑𝟗
𝟒𝟏
A₇₃=
𝟕𝟓
𝟕𝟕

24. Find the indicated term of each
sequence
1. an = 6n -
4 = 50
a₉
2. an = 1 -
𝟏
𝒏
=
𝟏𝟏
𝟏𝟐
a₁₂
3. an = (𝟒)𝒏−𝟑
( n +
2)
=
112
a₅
4. an = (−𝟐)𝒏−𝟓 =
256
a₁₃
5. an = 7n +
7 = 77
a₁₀

26. Sequence
A succession of numbers in a
specific order

27. Term
1, 2, 3, 4, 5
numbers in the sequence
1st term 5th / last term

28. Types of Sequences
Finite sequence
Infinite sequence

29. Finite Sequence
a sequence that ends
limited domain

30. Infinite Sequence
a sequence goes on forever
infinite domain

31. General terms of
sequences
nth term = function value is a(n) usually
denoted by an (read as “a sub n”)

32. The general term of a sequence is
given. Write the first four terms, the
10th term, and the 15th term
1. an = 2n -1
= 1, 3, 5, 7 ;
19; 29
2. an = 3n +
2
= 5, 8, 11,
14; 32; 47
3. an = n² + 2= 3, 6, 11, 18;
102;227
4. an = n² - 3
= -2, 1, 6, 13;
92 ; 222
5. an =
𝒏
𝒏+𝟏
=
𝟏
𝟐
,
𝟐
𝟑
,
𝟑
𝟒
,
𝟒
𝟓
;
𝟏𝟎
𝟏𝟏
;
𝟏𝟓
𝟏𝟔

33. The general term of a sequence is
given. Write the first four terms, the
10th term, and the 15th term
6. an =
𝒏
𝒏+𝟒
=
𝟏
𝟓
,
𝟏
𝟑
,
𝟑
𝟕
,
𝟏
𝟐
;
𝟓
𝟕
;
𝟏𝟓
𝟏𝟗
7. an = 1 +
𝟏
𝒏
= 2 ,
𝟑
𝟐
,
𝟒
𝟑
,
𝟓
𝟒
;
𝟏𝟏
𝟏𝟎
;
𝟏𝟔
𝟏𝟓

34. The general term of a sequence is
given. Write the first four terms, the
10th term, and the 15th term
8. an = 1 -
𝟏
𝒏
=0,
𝟏
𝟐
,
𝟐
𝟑
,
𝟑
𝟒
;
𝟗
𝟏𝟎
;
𝟏𝟒
𝟏𝟓
9. an =
𝒏 −𝟏
𝒏+𝟐
= 0 ,
𝟏
𝟒
,
𝟐
𝟓
,
𝟏
𝟐
;
𝟑
𝟒
;
𝟏𝟒
𝟏𝟕
10. an =
𝒏+𝟐
𝒏+𝟑
=
𝟑
𝟒
,
𝟒
𝟓
,
𝟓
𝟔
,
𝟔
𝟕
;
𝟏𝟐
𝟏𝟑
;
𝟏𝟕
𝟏𝟖

35. Find the indicated term of each
sequence
1. an = 8n -
18
= 78
a₁₂
2. an = 2n + 5 =33
a₁₄
3. an = (−𝟐)𝒏−𝟐
(2n +
1)
=
208
a₆
4. an = (1 -
𝟏
𝒏
) =
𝟗
𝟏𝟎
a₁₀

36. First Quarter
seatwork
Date Seatwork # & name Score
08 – 08 - 2022 Seatwork no. 1 – Sequence 25 / 25

38. Find the indicated term of each
sequence
1. an = 3n -
4
= 32
a₁₂
2. an = 6n + 5 =89
a₁₄
3. an = (−𝟑)𝒏−𝟏
(n +
1)
= -
1701
a₆
4. an = (1 -
𝟏
𝒏
)² =
𝟏𝟒
𝟏𝟓
a₁₅
5. an =
𝒏 −𝟑
𝒏+𝟐
=
𝟕
𝟏𝟐
a₁₀

40. Write formulas
for sequences

41. Recursive Form
The general term is expressed by using the
previous term.
2, 5, 8, 11, 14
𝑎𝑛 = 𝑎𝑛−1 + 3
𝑎6−1 + 3
𝑎6 =
= 𝑎5 + 3 = 17
= 14 + 3
6th 𝑡𝑒𝑟𝑚

42. Recursive Form
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The general term is expressed by using the
previous term.
18, 14, 10, 6, 2, -2
𝑎𝑛 = 𝑎𝑛−1 − 4

43. Recursive Form
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The general term is expressed by using the
previous term.
1, 3, 9, 27, 81
𝑎𝑛 = 𝑎𝑛−1 𝑥3
𝑎𝑛 = 3(𝑎𝑛−1)

44. Find for the Recursive Form.
1, 3, 7, 15, 31, 63
𝑎𝑛 = 2(𝑎𝑛−1) + 1
multiply previous term
by 2 add 1

45. Recursive Form
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The general term is expressed by using the
previous term.
3, 7, 11, 15, 19
𝑎𝑛 = 𝑎𝑛−1 +4

47. Explicit Form
The general term is expressed by using the position
of the term in the sequence.
Express the sequence into another
formula.
2, 4, 6, 8, 10, …
Express it through a table first.
Term (𝒂𝒏) 2 4 6 8 10
Position (𝑛) 1
(1st)
2
(2nd)
3 4 5

48. Use the position in getting the formula of the general term.
Term (𝒂𝒏) 2 4 6 8 10
Position (𝑛) 1
(1st)
2
(2nd )
3
(3rd )
4
(4th)
5
(5th)
𝑎𝑛 = (𝑛)
2
Explicit Form
To get 2, multiply the first term (1) by 2
To get 4, multiply the second term (2) by 2
To get 6, multiply the third term (3) by 2

49. Explicit Form
3, 6, 9, 12, 15, 18
𝑎𝑛 = 3𝑛
Term (𝒂𝒏) 3 6 9 12 15
Position (𝑛) 1 2 3 4 5

50. Explicit Form
The general term is expressed by using the position
of the term in the sequence.
5, 10, 15, 20
𝑎𝑛 = 5𝑛

51. 1, 4, 9, 16, 25,…
𝑎𝑛 = n²
Term (𝒂𝒏) 1 4 9 16 25
Position (𝑛) 1 2 3 4 5

52. 2, 5, 10, 17, 26,…
𝑎𝑛 = n² + 1
Term (𝒂𝒏) 2 5 10 17 26
Position (𝑛) 1 2 3 4 5
𝑎𝑛 = 2n
𝑎𝑛 = 2n + 1
𝑎𝑛 = n²
𝑎𝑛 = n² + 1

53. 1, 4, 7, 10, 13,…
𝑎𝑛 = 3n − 2
Term (𝒂𝒏) 1 4 7 10 13
Position (𝑛) 1 2 3 4 5
𝑎𝑛 = n²
𝑎𝑛 = 2n − 1
𝑎𝑛 = 3n
𝑎𝑛 = 3n – 2

54. Explicit Form
10, 15, 20, 25, 30, …
𝑎𝑛 = 5(𝑛 + 1)

55. Find the Explicit Form.
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12, 16, 20, 24, 28, …
𝑎𝑛 = 4(𝑛 + 2)

56. Find a possible formula for the general terms
of the sequence
a. 1, 8, 27, 64, 125,… an = n³
b. 2, 4, 6, 8, 10, 12,… an = 2n
c. 1, 3, 5, 7, 9,11,… an = 2n - 1

58. Fibonacci
Sequence
The next number is obtained by
adding the previous two terms
Fibonacci Pattern

59. What are the next three
numbers after the
sequence?
1, 1, 2, 3, 5, 8, …
13 21 34
+ + + + +

60. Leonardo
Pisano Bigollo
-born in Pisa, Italy
-a famous Italian Mathematician
of the Medieval Period
-famous for his work, Liber Abaci Leonardo Pisano Bigollo

61. Fibonacci
Pattern
-it is a pattern used in taking
photos in a great manner
Fibonacci Pattern
-uses a proper
concentration of the
foreground against the
background

62. Fibonacci
Pattern

63. Fibonacci
Pattern
Golden ratio

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65. Learn Mathematics online, the best ways we can

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70. chsths.aralinks.net
 My Subjects
 Math 10 Faith
 First Quarter
 Seatwork no. 1
(follow the instruction)