geometric sequences

1. 11.3 – Geometric11.3 – Geometric
SequencesSequences

2. What is a Geometric Sequence?What is a Geometric Sequence?
 In a geometric sequence, the ratioIn a geometric sequence, the ratio
between consecutive terms isbetween consecutive terms is
constant. This ratio is called theconstant. This ratio is called the
common ratiocommon ratio..
 Unlike in an arithmetic sequence, theUnlike in an arithmetic sequence, the
difference between consecutivedifference between consecutive
terms varies.terms varies.
 We look forWe look for multiplicationmultiplication toto
identify geometric sequences.identify geometric sequences.

3. Ex: Determine if the sequence is geometric.Ex: Determine if the sequence is geometric.
If so, identify the common ratioIf so, identify the common ratio
 1, -6, 36, -2161, -6, 36, -216
yes. Common ratio=-6yes. Common ratio=-6
 2, 4, 6, 82, 4, 6, 8
no. No common rationo. No common ratio

4. Important Formulas forImportant Formulas for
Geometric Sequence:Geometric Sequence:
 Recursive FormulaRecursive Formula  Explicit FormulaExplicit Formula
an = (an – 1 ) r an = a1 * r n-1
Where:
an is the nth term in the
sequence
a1 is the first term
n is the number of the term
r is the common ratio
 Geometric MeanGeometric Mean
Find the product of
the two values and
then take the square
root of the answer.

5. Let’s start with the geometric meanLet’s start with the geometric mean
 Find the geometric mean betweenFind the geometric mean between
3 and 483 and 48
Let’s try one: Find the geometric mean
between 28 and 5103

6. Ex: Write the explicit formula forEx: Write the explicit formula for
each sequenceeach sequence
First term: aFirst term: a11 = 7= 7
Common ratio = 1/3Common ratio = 1/3
Explicit: an = a1 * r n-1
Now find the first five
terms:
a1 = 7(1/3) (1-1)
= 7
a2 = 7(1/3) (2-1)
= 7/3
a3 = 7(1/3) (3-1)
= 7/9
a4 = 7(1/3) (4-1)
= 7/27
a5 = 7(1/3) (5-1)
= 7/81

7. Explicit Arithmetic Sequence ProblemExplicit Arithmetic Sequence Problem
Find the 19Find the 19thth
term in the sequence ofterm in the sequence of
11,33,99,297 . . .11,33,99,297 . . .
a19 = 11(3)18
=4,261,626,379
Common ratio = 3
a19 = 11 (3) (19-1)
Start with the explicit sequence formula
Find the common ratio
between the values.
Plug in known values
Simplify
an = a1 * r n-1

8. Let’s try oneLet’s try one
Find the 10Find the 10thth
term in the sequence ofterm in the sequence of
1, -6, 36, -216 . . .1, -6, 36, -216 . . .
a10 = 1(-6)9
= -10,077,696
Common ratio = -6
a10 = 1 (-6) (10-1)
Start with the explicit sequence formula
Find the common ratio
between the values.
Plug in known values
Simplify
an = a1 * r n-1