Geometric Sequences (with some arithmetic sprinkled in)
Geometric Sequences – we multiply/divide to get the
next value.
1.) Given the geometric sequence, write down
what the common ratio (r) is.
a.) 5, 10, 20, 40, ...
r=
b.) -11, 22, -44, 88, ...
r=
c.) 100, 50, 25, 12.5, …
r=
1 1
d.) 8, 4, 2, 1, ,
2 4
r=
3.) Find the 10th number in this sequence:
1, 3, 9, 27, 81, …
4.) Find the 8th number in this sequence:
-10, 20, -40, 80, ...
5.) Write the first five numbers in the sequence that
represents the following explicit functions:
a.) f(x) = 2x + 3
b.) f(x) = 2x
Arithmetic Sequences
Geometric Sequences
You add or subtract to
get the next numbers
You multiply or divide
to get the next numbers
They have a common
difference, (d)
They have a common
ratio, (r)
They can be written
explicitly using a linear
equation:
y  mx  b
They can be written
explicitly using an
exponential equation:
y  y n (ratio) x  xn  k
2.) Are the following sequences arithmetic,
geometric, or neither?
a.) 3, 6, 9, 12, 15, …
Match the sequence with the explicit function.
6.) 1, 2, 3, 4, 5…
a.) f(x) = 2x
7.) 6, 7, 8, 9, 10…
b.) f(x) =
8.) 5, 10, 20, 40, …
c.) f(x) = x + 5
9.) 5, 25, 125, 675,…
d.) f(x) = x
5
2
 5x
10.) Write the first five numbers in the sequence that
represents the following recursive functions:
a.) a1 = 8, an = an-1 + 2
b.) a1 = 3, an = 4an-1
b.) 7, 14, 28, 56, 112, …
Match the sequence with the recursive function.
c.) 5, -10, 20, -40, 80, -160, …
11.) 6, 12, 24, 48, 96
12.) 6, 7, 8, 9, 10…
d.) 4, 8, 12, 24, 48, 50, …
a.) a1 = 6, an =
1
2
(an-1)
b.) a1 = 6, an = an-1 + 2
13.) 6, 8, 10, 12, 14
c.) a1 = 6, an = 2(an-1)
14.) 6, 3, 1.5, .75, .375
d.) a1 = 6, an = an-1 + 1
Geometric Sequences
defined explicitly
15.) 4, 20, 100, 500, 2500, ...
a.) What is the common ratio?
b.) What is a 0 ?
c.) What is the explicit equation?
Use a’s and n’s
y  a 0 (ratio) x
Geometric Sequences
defined recursively
To define a function recursively:
1.) always give people the 1st term (a1)
2.) Tell people what to do to get the next
number using the previous number.
18.)
4, 12, 36, 108, ...
a1 =
an = (
19.)
16.)
90, 30, 10,
)an-1
800, 400, 200, 100, 50, …
a1 =
10
, ...
3
an = (
)an-1
a.) What is the common ratio?
b.) What is a 0 ?
c.) What is the explicit equation?
Use a’s and n’s
y  a 0 (ratio) x
20.)
4, 20, 100, 500, 2500, ...
a1 =
an =
17.)
10
, ...
3
21.)
90, 30, 10,
22.)
12, 7, 2, -3, -8, -13,…
22, -44, 88, -176...
a.) What is the common ratio?
b.) What is a 0 ?
c.) What is the explicit equation?
Use a’s and n’s
y  a 0 (ratio) x
23.) 22, -44, 88, -176...
Recursive and Explicit
1.) 7, 14, 28, 56, 112, ...
a64 =
a.) Write the explicit equation that represents the
sequence:
b.) Write the recursive equation that represents the
sequence:
2.) 320, 80, 20, 5, 1.25, ...
a64 =
a.) Write the explicit equation that represents the
sequence:
b.) Write the recursive equation that represents the
sequence:
3.) 6, -12, 24, -48, 96...
a64 =
a.) Write the explicit equation that represents the
sequence:
b.) Write the recursive equation that represents the
sequence:
4.) 6, 21, 73.5, 257.25, 900.375, ...
a64 =
a.) Write the explicit equation that represents the
sequence:
b.) Write the recursive equation that represents the
sequence:
5.) 4, 2, 1,
a64 =
1
2
,
1
4
a.) Write the explicit equation that represents the
sequence:
b.) Write the recursive equation that represents the
sequence:
6.) 5, 7, 9, 11, 13, 15, 17…
a64 =
a.) Write the explicit equation that represents the
sequence:
b.) Write the recursive equation that represents the
sequence:
7.)
At the beginning
At one minute
At two minutes
a.) Write the explicit equation that represents
the sequence:
At three minutes
At four minutes
b.) Write the recursive equation that represents the sequence:
8.)
a.) Write the explicit equation that represents the
sequence:
b.) Write the recursive equation that represents the
sequence:
Given the recursive definition, write
the explicit definition.
Given the explicit definition, write the
recursive definition.
9.) a1  4
12.) a n  5(2) n
a n  3  a n1
10.) a1  5
a n  2  a n 1
11.) a1  10
an  an1  2
13.)
1
a n  200 
2
14.) an  3n  1
n