The document discusses arithmetic sequences and series, including how to determine if a sequence is arithmetic, how to find subsequent terms using the common difference, and how to calculate the sum of the first n terms of an arithmetic series using the arithmetic series formula. Examples are provided to illustrate how to identify arithmetic sequences, calculate missing terms, find common differences, and compute sums of arithmetic series.

1. Morning Drill
Evaluate the following:
1. y ÷2 + x ; x=1 and y=2
2. p2 + m ; m=1 and p=5
3. m + p ÷ 5 ; m=1 and p=5
4. z(x + y) ; x=6, y=8 and z=6
5. p3 + 10 + m ; m=9 and p=3

2. ARITHMETIC
SEQUENCE
Mr. Zaint Harbi A. Habal
Teacher

3. ARITHMETIC SEQUENCE
- Is a sequence where each succeeding
term is obtained by adding a fixed number.
The fixed number is called the common
difference which is denoted as d.

4. What’s new
Determine whether the sequence is
arithmetic or not. If it is, find the common
difference.
1. 2, 6, 10, 14, . . .
2. –4, 8, –16, 32, –64, . . .
3. 2, 1, 1/2, 1/4, 1/8, . . .
4. 20, 13, 6, –1, –8, . . .
5. 2, 2 1/2 , 3, 3 1/2, . . .

5. What I know?
Given by the arithmetic sequence 3, 7, 11, 15, …
a. What is the common difference?
b. How many terms will you find if you are going to
evaluate a15?
c. What is the value of the 1st term? 2nd term? 3rd
term? 4th term?
d. If the sequence will be continued, what do you
think is the value of the tenth term? 15th term? 20th

6. Think-Pair-Share
Example 1
Consider the sequence 3, 7, 11, 15, . . . , what is the
15th term in the given sequence?
Example 2
What is the 15th term of the sequence
5, 3, 1, –1, –3, –5, . . .?
Example 3
In the arithmetic sequence -3, 0, 3, 6, …, which term
is equal to 138?

7. ARITHMETIC SEQUENCE
To find the next terms in an arithmetic
sequence, we use the formula:
𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
Where;
an – the last nth term
a1 – the first term
n – the number of terms in the
sequence

8. Study the given examples below and
identify if it is arithmetic or not.
1. 10, 13, 16, 19, …
2. 2, 6, 18, 54, …
3. 57, 49, 41

9. Example 1:
Determine the 10th term in the
sequence 4, 6, 8, 10, …
𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
The first element / term: a1 = 4.
The common difference: d = 2
The term: n = 10

10. Example 2:
Find the 7th term of an arithmetic
sequence given the first three terms 2, 6,
10.
Example 3:
Find the 10th term of an arithmetic
sequence given the first 4 terms 10, 19,
28, 37.

11. Seat Work / Assignment:
1. Find the 12th term of an arithmetic
sequence whose first term is 38 and
common difference of – 2.
2. Find the 15th term of the arithmetic
sequence 2, 4, 6.
3. Find the 25th term of the arithmetic
sequence 13, 16, 19, 22, …

12. Example 4:
Form an arithmetic sequence with 1st
term 3 and 7th term 15.
Example 5:
If the 6th term of an arithmetic sequence
is 24 and the 12th term is 48, find the first
term.

13. Hint: Find the common difference using the
formula:
𝒅 =
𝒂𝒏−𝒂𝒎
𝒏−𝒎
where
am is the first given term
an is the last given term
m is the position of am
n is the position of an

14. Example 6:
If a38 = 140 and a51 = 192, what is a5?
Example 7:
What is the first 5 terms of an arithmetic
sequence whose 23rd term is –107 and
whose 55th term is –267?

15. What I know?
Guess the missing term on the following
sequence then find the sum.
1. -1, ___, ___, ____, ___, 14
2. 14, ___, ___, ___, ___, ___, 86
The missing number/s is/are called the
arithmetic mean/s of the two numbers.

16. ARITHMETIC MEAN
- it is the terms between any two
nonconsecutive terms of an arithmetic
sequence.

17. ARITHMETIC MEAN
Illustrative Example:
Find two arithmetic means between 2 and
8.
*Using d = 2, generate the next terms by
adding “d” to the previous term.*
So a2 = a1 + d and a3 = a2 + d which
means

18. ARITHMETIC MEAN
You may use the formula for the
common difference to find the arithmetic
mean.
𝒅 =
𝒂𝒏−𝒂𝟏
𝒏−𝟏
or 𝒅 =
𝒂𝒏−𝒂𝒎
𝒏−𝒎

19. Going back to What I know?
Example 1:
Guess the missing term on the following
sequence then find the sum.
1. -1, ___, ___, ____, ___, 14
2. 14, ___, ___, ___, ___, ___, 86

20. Example:
2. Find the arithmetic mean of 7 and
15.
3. Find the four arithmetic means
between 7 and -13.

21. Example 4:
Find the sum of the first:
a. five positive numbers
b. ten positive numbers
c. 20 positive numbers
d. 100 positive even numbers

22. Example 5:
Find the sum of the first 20 terms of
an arithmetic sequence 2, 5, 8, 11, …

23. ARITHMETIC SERIES
Arithmetic series is an indicated sum of
the first n terms of an arithmetic sequence.
The sum of n terms is denoted by Sn.

24. ARITHMETIC SERIES
The formula in finding arithmetic series is
𝑺𝒏 =
𝒏
𝟐
(𝒂𝟏 + 𝒂𝒏)
𝑺𝒏 =
𝒏
𝟐
[𝟐𝒂𝟏 + 𝒏 − 𝟏 𝒅]

25. Example 4:
Find the sum of the first:
a. five positive numbers
b. ten positive numbers
c. 20 positive numbers
d. 100 positive even numbers

26. Example 5:
Find the sum of the first 20 terms of
an arithmetic sequence 2, 5, 8, 11, …
Example 6:
Find the sum of the first 10 terms of
the arithmetic sequence 4, 10, 16, 22, 28,
…

27. Example 7:
Find the sum of the first 30 multiples of 5.
Example 8:
Find the sum of the first 25 multiples of 3

28. Assignment:
1. If a38 = 140 and a51 = 192, what is a1?
2. What are the two arithmetic means of the terms -8 and
100?
3. What is the sum of the first 24 terms of the arithmetic
sequence: 4,8,12,16,…?
4. Find the sum of all odd integers from 10 to 100.
5. What is the 10th term of the arithmetic sequence -4, 1, 6,
11, …?