Warm-Up
1. Grab a worksheet off the
stool.
2. Complete all of the odds.
Homework Check
Geometric Sequences
and Series
What is a geometric sequence?
• In a geometric sequence, each term is found by
multiplying the previous term by a constant.
What is a geometric sequence?
In general you write a Geometric Sequence like this:
{a, ar, ar2, ar3, ... }
Where a is the first term, and r is the factor
between the terms called the common ratio.
Example:
a=1
r=2
Term
#
How it is written with “r”
1st
1
1
2nd
2
1•2
3rd
4
1•22
4th
8
1•23
What do you notice?
• Do you see a pattern between term #
and the exponent r is raised to?
Term
#
How it is written with “r”
1st
2
2
2nd
4
2•21
3rd
8
2•22
4th
16
2•23
5th
32
2•24
6th
64
2•25
• One less than the term number!
Formula
an = a1
(n-1)
r
Where an is the term we are looking for
and a1 is the first term in the
sequence…so if we want to find the
third term (a3) we plug in 3 for n!
Look how we find each term…
Sequence
Term
Formula
a1r(n-1)
Final Result
2
a1
a1r(1-1) = 2•2(1-1)
2•20 = 2•1 = 2
4
a2
a1r(2-1) = 2•2(2-1)
2•21 = 2•2 = 4
8
a3
a1r(3-1) = 2•2(3-1)
2•22 = 2•4 = 8
16
a4
a1r(4-1) = 2•2(4-1)
2•23 = 2•8 = 16
32
a5
a1r(5-1) = 2•2(5-1)
2•24 = 2•16 = 32
64
a6
a1r(6-1) = 2•2(6-1)
2•25 = 2•32 = 64
The graph….
• We graph the order of the term for “x”
and the number in that spot for “y”
Term Sequence
X
Y
Point
1st
1
(1,1)
2nd
2
(2,2)
3rd
4
(3,4)
4th
8
(4,8)
5th
16
(5,16)
Examples
1) What is the first term,
a1? What is the common
ratio? Find the next 3
terms….
2) What is the first term,
a1? What is the common
ratio? Find the next 3
terms….
1, 3, 27, ___ , ___ , ___
70, 7, 0.7, ____ , ____ , ____
You Try
1) What is the first term,
a1? What is the common
ratio? Find the next 3
terms….
2) What is the first term,
a1? What is the common
ratio? Find the next 2
terms….
-3 , -6, -12, ___ , ___ , ___
8, -20, 50, -125, ____ , ____
Examples
3) Set up a geometric
sequence with a1 = 5, and
the common ratio of 2.
Find the 4th term.
4) Set up a geometric
sequence with a1 = 7 , and
the common ratio of 1/5 .
Find the 3rd term.
You Try
3) Set up a geometric sequence
with a1 = 6, and the common
ratio of 3. Find the 4th term
using your formula.
4) Set up a geometric sequence
with a1 = 2 , and the common
ratio of ¼ . Find the 3rd term
using your formula.
Examples
5) Find the first, fourth, and
eighth term of each sequence.
6) Find the first, third, and sixth
an = 4•2(n-1)
an = (0.5)•3(n-1)
term of each sequence.
You Try
5) Find the first, fourth, and
eighth term of each
sequence.
6) Find the first, fifth, and
an = -2 • 5(n-1)
an = 0.25 • 3(n-1)
seventh term of each
sequence.
Decide if each sequence is geometric or
arithmetic….
Sum of A Finite Geometric
Series
• Finite- having limits, something that
is measureable.
• Finite Series- the sum of the terms
of a sequence.
• Geometric series-series in which the
ratio of each two consecutive terms is
a constant function of the sum.
Sum of A Finite Geometric
Series
• Formula:
a1 (1  r )
Sn 
1 r
n
a1  first term n  number of terms
r  common ratio
Example:
1. What is the sum of the first ten
terms of the geometric series?
8 + 6 + 32 + 64 + 128 + ….
Example:
-10 – 5 – 2.5 – 1.25 - … ; n = 7
You Try:
3 + 12 + 48 + 192 + …; n = 6
Homework
• Workbook pg. 13 #1-9
Extra Credit:
• Independent Practice Sheet:
Evens, on separate sheet of paper.
• Due Friday March 20.