Algebra II – Ch. 9 Day #4
Topics: Arithmetic and Geometric Series
STANDARDS/GOALS:
F.IF.3/H.2.e

I can define a finite
arithmetic series and find
the sum of the series.
A.SSE.4/H.2.c

I can define a finite or
infinite geometric series
and find the sum of the
series.
Notes and Definitions:
The numbers in a sequence are called terms.
A sequence is a pattern of consecutive terms separated with a
comma.
A finite sequence ends, while an infinite sequence continues without
stopping.
A series is a pattern of consecutive terms similar to a sequence. In a
series, the terms are separated by a + sign and we add up all of the
terms.
ARITHMETIC SERIES:
The rule for an arithmetic series is the same as the rule for an
arithmetic sequence. The only difference is that we can get a sum for
the arithmetic series.
𝐚𝐧 = 𝐚𝟏+𝐝(𝐧−𝟏)
Example 1: Finite Sequence
3,5,7,9
Example 2: Infinite Sequence
3,5,7,9,……….
Example 3: Finite Series
3+5+7+9
Example 4: Infinite Series
3+5+7+9+………
The explicit formula for this sequence and series would be
𝒂𝒏 = 𝟑𝒏 + 𝟏
SUMMATION NOTATION:
Summation Notation is used to write a series.
The Greek letter, sigma, means “to sum”.
When summation notation is used the number
on the bottom of the symbol is the starting term,
the number on the top of the symbol is the
ending term.
Example 5: Summation Notation
5
∑ 2n + 1
𝑛=1
This means that I will substitute 1,2,3,4,and 5
into the rule and add the five terms together.
2(1)+1=3 2(2) + 1 = 5
2(3) + 1 = 7
2(4) + 1 = 9 2(5) + 1 =11
So the series expanded out would look like
3+5+7+9+11. The sum would be 35.
Sums of Finite Arithmetic Series:
𝒔
𝒏=
𝒏(𝒂𝟏 +𝒂𝒏)
𝟐
where n is the number of terms, 𝒂𝟏 is the first term, and 𝒂𝒏 is
the last term in the series.
Example 6: Use the formula to find the sum of the
arithmetic series, 7+9+11+13+15+17
𝟔(𝟕 + 𝟏𝟕)
𝒔𝒏 =
= 𝟕𝟐
𝟐
Geometric Series:
Geometric Series are similar to geometric sequence. The
explicit formula is the same, which is 𝒂𝒏 = 𝒂𝟏 ∗ 𝒓𝒏−𝟏 .
Example 7: 3+9+27+81
This is a geometric series whose explicit formula would be
𝒂𝒏 = 𝟑 ∗ 𝟑𝒏−𝟏
Geometric Series in Summation Notation:
5
∑ 2 ∗ 3n−1
𝑛=1
2 + 6 + 18 + 54 + 162
Sum of a Geometric Series:
Example 8:
2 + 6 + 18 + 54 + 162
𝒔𝒏= 𝒂𝟏(𝟏−𝒓𝒏)
𝟏−𝒓
𝒔𝒏
𝟐(𝟏−𝟑𝟓 )
=
−𝟐
= 242
Sums of Infinite Geometric Series:
𝒂
𝟏
S = 𝟏−𝒓
This only occurs if the absolute value of the ratio( r) is less
than 1.
Example 9:
½ + ¼ + …..
S=
𝟏
𝟐
𝟏−
𝟏
𝟐
=𝟏
Homework Ch. 9 Day #4 ( arithmetic and geometric series)
Write each arithmetic series into summation notation and
find the sum of each arithmetic series in numbers 1-10, using
the formula
𝒔 𝒏(𝒂𝟏 +𝒂𝒏)
𝒏=
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
𝟐
-3+-6+-9+-12
7+9+11+13+15+17
1+2+3
4+13+22+31+40
7+10+13+16+19+22+25
8+16+24+32+40
-4+-9+-14+-19
4+10+16+22+28+34+40+46+52
10+20+30+40
-5+-10+-15+-20
Write each geometric series in summation notation and find
the sum of each geometric series.
1. 1, 2, 4, 8
2. ½, ¼,
𝟏
𝟖
3. 3, 6, 12, 24
4.
5.
𝟏
,
𝟑
𝟐
𝟏
,
𝟏𝟐 𝟒𝟖
𝟒
, ,
𝟑
𝟏
𝟖
𝟗 𝟐𝟕
6. 8, 32, 128
7. 5, 25, 125
8. 10, 30, 90
9. 10, 100, 1000
10.
𝟏
𝟓
,
𝟏
,
𝟏
𝟏𝟎 𝟐𝟎
Find the sum of the infinite geometric series, if it
exists.
1. 1, 2, 4, 8, …..
𝟏
2. ½, ¼, , ….
𝟖
3. 3, 6, 12, 24 , …
4.
5.
𝟏
𝟑
𝟐
𝟑
,
𝟏
,
𝟏
𝟏𝟐 𝟒𝟖
𝟒
, ,
𝟖
𝟗 𝟐𝟕
, …..
, ……