arithmetic mean & arithmetic series
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ARITHMETIC MEANARITHMETIC SERIES
ARITHMETIC MEAN
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
ARITHMETIC MEAN
Arithmetic Mean =
Formula
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
ARITHMETIC MEAN
•Use a6=a1+5d to solve for d.
25= 5+5d
5d=25-5
5d=20
d=4
Example 1
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
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
ARITHMETIC MEAN
Use a6=a1+5d to solve for d.
-4= -24+5d
5d=4-24
5d=-20
d=-4
Example 2
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
ARITHMETIC SERIES
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.
ARITHMETIC SERIES
Sn = (a1 + an)
Formulae
Sn = [2a1 + (n - 1) d]
ARITHMETIC SERIES
3 + 8 + 13 + … + 73
• a1 = 3
• an = 73
• d = 5• n = ?
• Sn = ?
Example 1
ARITHMETIC SERIES
Sn = (a1 + an)
S15 = (3 + 73)
S15 = (76)
S15 = (38)
S15 =
Example 1
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
ARITHMETIC SERIES
an = a1 + (n – 1) d
a10 = 3 + (10 – 1) 4
a10 = 3 + (9) 4
a10 = 3 + 36
a10 = 39
Example 2
ARITHMETIC SERIES
Sn = (a1 + an)
S10 = (3 + 39)
S10 = 5 (42)
S10 =
Example 2
ARITHMETIC MEANARITHMETIC SERIES