Arithmetic Series

1. Arithmetic
Series

2. PRAYER

3. OBJECTIVES
1. Know the difference between arithmetic sequence and series.
2. Determine the sum of the first n terms of an arithmetic sequence.
3. Apply the concept of arithmetic series in real – life situations.

4. Sequence vs Series

5. Sequence vs Series
Sequence is a list of elements that follow a special pattern.
Series is the sum of the terms of a sequence.
5, 9, 13, 17, …
5 + 9 + 13 + 17 +… + n

6. LET’S DISCOVER!
What is the sum of the first 100 natural numbers?
1+2+3+4+…+97+98+99+100
101
Is there a pattern that may help you in finding the sum?

7. LET’S DISCOVER!
1+2+3+4+…+97+98+99+100
101
What is the sum of each pair of numbers?
How many pairs have this sum?
What is the sum of the first 100 natural numbers?
101
50
5050

8. TRY THIS!
What is the sum of the first 100 natural numbers?
1+2+3+4+…+97+98+99+100
5050

9. LET’S DISCOVER!
1+2+3+4+…+97+98+99+100
101
Sum of each pair:
Number of pairs:
The sum of the first n terms:
101
50
5050
𝐴1 + 𝐴𝑛
𝑛
2
𝑆𝑛 =
𝑛
2
(𝐴1 + 𝐴𝑛)

10. DO YOU KNOW?
Karl Friedrich Gauss
answered the same
question when he was in
primary school.

11. EXAMPLE 1
Find the sum of the first 10 terms of an arithmetic
sequence whose first term is 3 and 10th term is 66.
10
66
A10:
S10
Sn =
n
2
(A1 + An)
S10 =
10
2
(3 + 66)
S10 = 345
A1:
An:
n:
Sn:
3

12. EXAMPLE 2
Find S16 for the arithmetic sequence whose first term
is – 4 and 16th term is 71.
A1:
A16:
n:
Sn:
– 4
16
71
S16
Sn =
n
2
(A1 + An)
S16 =
16
2
(−4 + 71)
S16 = 536

13. LET’S ANSWER!
Find S20 for the arithmetic sequence whose first term
is 11 and 20th term is – 65.
A20:
Sn:
11
20
– 65
S20
Sn =
n
2
(A1 + An)
S20 =
20
2
[11 + −65 ]
S20 = −540
A1:
n:

14. LET’S DISCOVER
What is the sum of the first 30 terms of an arithmetic
sequence whose first term is – 8 and common difference is 7.
A1:
A30:
n:
Sn:
– 8
30
?
S30
Sn =
n
2
(A1 + An)
S30 =
30
2
(−8 + 195)
S30 = 2805
An = A1 + n − 1 d
A30 = −8 + 30 − 1 (7)
A30 = 195

15. LET’S DISCOVER
Sn =
n
2
(A1 + An)
An = A1 + n − 1 d
Sn =
n
2
[A1 + A1 + n − 1 d]
Sn =
n
2
[2A1 + (n − 1)(d)]
Formula in finding
the sum of the first n
terms if the last term
is not given.

16. LET’S ANSWER!
What is the sum of the first 18 terms of an arithmetic
sequence whose first term is 9 and common difference is 5?
A1:
A18:
n:
Sn:
9
18
?
S18
Sn =
n
2
[2A1 + n − 1 d]
S18 =
18
2
[2 9 + 18 − 1 5 ]
S18 = 927

17. LET’S ANSWER!
The fifth term of an arithmetic sequence is 39 and the common
difference is 10, what is the sum of the first 12 terms of the
sequence?
A1:
d:
n:
Sn:
39
12
10
S12
Sn =
n
2
[2A1 + n − 1 d]
S12 =
12
2
[2 −1 + 12 − 1 10 ]
S12 = 648
A1: - 1

18. DO YOU KNOW?
“IPON”
CHALLENGE

19. LET’S ANSWER!
If you are going to start saving money with Php 5 this week , Php 7
pesos on the second week, and Php 9 on the third week, how much
will you be able to save in total on the 10th week?
A1:
d:
n:
Sn:
5
10
2
S10
Sn =
n
2
[2A1 + n − 1 d]
S10 =
10
2
[2 5 + 10 − 1 2 ]
S10 = 140

20. LET’S SUM IT UP!
What is the difference between arithmetic
sequence and arithmetic series?
What are the two formulas we use in finding
for the sum of the first n terms of an
arithmetic sequence?

21. ANSWER!
Exercises on your Self Learning Module in
Mathematics 10