Arithmetic Series
Summing or adding the terms of an arithmetic sequence creates what is
called a series.
Sequence: 2, 4, 6, 8, 10, 12
(list)
Series: 2 + 4 + 6 + 8 + 10 + 12 (sum)
Formulas to find the nth Partial Sum of an Arithmetic Sequence.
𝒏
Sn = (a1 + an)
𝟐
on your formula card
Let’s see if we can figure out where these formulas come from.
1 + 2 + 3 + 4 + 5 + … + 96 + 97 + 98 + 99 + 100
Skip explanation of second formula, we do not use it!
Practice: Determine the sum of the arithmetic series.
1.
3 + 8 + 13 + … + 73
2.
an = -4n + 3; n = 20
Geometric Series
Summing or adding the terms of a geometric sequence creates what is
called a series.
Sequence: 2, 4, 8, 16, 32, 64
(list)
Series: 2 + 4 + 8 + 16 + 32 + 64
(sum)
Partial Sum of first nth terms of a Geometric Sequence.
Sn =
𝒂𝟏(𝟏−𝒓𝒏 )
𝟏−𝒓
on your formula card
Practice: Determine the sum of the geometric series.
1.
3 + 6 + 12 + … + 1536
2.
an = 2(-3)n-1; n = 5
Summation Notation
Greek letter called Sigma.
It’s a compact way to write a sum.
where to “end” the sum
𝑛
∑ 𝑎𝑖
𝑖=1
index is a variable/letter
where to “start” the sum
Examples:
4
∑ 𝑖2
𝑖=0
4
∑ 𝑖2
𝑖=2
7
1
∑(2𝑖 + )
𝑖
𝑖=4
5
∑𝑖
Summation can be done on
the calculator!
If it’s multiple choice, just
plug it in.
If it’s show your work, just
show substitutions, then do
it calculator to get
summation.
On a calculator page, catalog
(below del button), #4 (must
push 4 button), is the
middle of second row is
sigma symbol, now fill in the
parts and press enter.
𝑖=1
5
∑𝑖
𝑖=1
Rewrite using summation notation:
3 + 6 + 9 + 12 + 15 + 18 + 21
26.5 + 24.5 + .22.5 + …+ 2.5
Writing a Geometric Series using Sigma / Summation
Notation
Express the sum using sigma/summation notation:
1 + .1 + .01 + .001 + .0001 + …
Arithmetic Series in Sigma Notation
Write the following in sigma notation.
75 + 70 + 65 + … + (-25)