The document defines arithmetic mean and arithmetic series. Arithmetic mean is calculated by adding all terms of a sequence and dividing by the total number of terms. It provides examples of inserting arithmetic means between given terms. An arithmetic series is the sum of an arithmetic sequence. The sum is calculated by multiplying the number of terms by the average of the first and last terms. It provides examples of calculating the sum of arithmetic series.

1. ARITHMETIC MEAN
ARITHMETIC SERIES

2. ARITHMETIC MEAN

3. ARITHMETIC MEAN
Finding a certain number of terms between two given
terms of an arithmetic sequence
The terms between any two nonconsecutive terms of
an arithmetic sequence are known as arithmetic
means

4. ARITHMETIC MEAN
Arithmetic Mean =
𝑥1
+ 𝑥2
+ 𝑥3
+ … + 𝑥𝑛
𝑛
Formula

5. ARITHMETIC MEAN
Insert 4 arithmetic means between 5 and 25.
• Since we are required to insert 4 terms, then there will be 6 terms in all.
• Let a1=5 and a6=25, we will insert a2, a3, a4, a5 as shown below.
5, a2, a3, a4, a5, 25
Example 1

6. ARITHMETIC MEAN
•Use a6=a1+5d to solve for d.
25= 5+5d
5d=25-5
5d=20
d=4
Example 1

7. ARITHMETIC MEAN
Using the value of d, we can now get the values of a2, a3, a4, a5, thus:
a2=9, a3=13, a4=17, a5=21
Example 1
5, 9, 13, 17, 21, 25

8. ARITHMETIC MEAN
Insert 4 arithmetic means between -24 and -4.
• Since we are required to insert 4 terms, then there will be 6 terms in all.
• Let a1=-24 and a6=-4, we will insert a2, a3, a4, a5 as shown below.
-24, a2, a3, a4, a5, -4
Example 2

9. ARITHMETIC MEAN
Use a6=a1+5d to solve for d.
-4= -24+5d
5d=4-24
5d=-20
d=-4
Example 2

10. ARITHMETIC MEAN
Using the value of d, we can now get the values of a2, a3, a4, a5, thus:
a2=-20, a3=-16, a4=-12, a5=-8
Example 2
-24, -20, -16, -12, -8, -4

11. ARITHMETIC SERIES

12. ARITHMETIC SERIES
It is the sum of an arithmetic sequence
The sum of an arithmetic series is found by multiplying
the number of terms times the average of the first and
last terms.

13. ARITHMETIC SERIES
Sn =
𝑛
2
(a1 + an)
Formulae
Sn =
𝑛
2
[2a1 + (n - 1) d]

14. ARITHMETIC SERIES
3 + 8 + 13 + … + 73
• a1 = 3
• an = 73
• d = 5
• n = ?
• Sn = ?
Example 1

15. ARITHMETIC SERIES
Sn =
𝑛
2
(a1 + an)
S15 =
15
2
(3 + 73)
S15 =
15
2
(76)
S15 = 15 (38)
S15 = 490
Example 1

16. ARITHMETIC SERIES
What is the sum of the first 10 terms in the series:
3, 7, 11, …
• a1 = 3
•d = 4
• n = 10
• an = ?
• Sn = ?
Example 2

17. ARITHMETIC SERIES
an = a1 + (n – 1) d
a10 = 3 + (10 – 1) 4
a10 = 3 + (9) 4
a10 = 3 + 36
a10 = 39
Example 2

18. ARITHMETIC SERIES
Sn =
𝑛
2
(a1 + an)
S10 =
10
2
(3 + 39)
S10 = 5 (42)
S10 = 210
Example 2

19. ARITHMETIC MEAN
ARITHMETIC SERIES