Ch2 O: Recursive & Explicit Equations for Arithmetic & Geometric Sequences
Name: ___________________________ Per. _____
A1.7.C Express arithmetic and geometric sequences in both explicit & recursive forms (& use the forms to find specific terms in the sequence)
Study Notes:
Arithmetic Sequence: 7, 10, 13, …
Geometric Sequence:
Add or subtract to get to the next term
Multiply or divide to get to the next term
Recursive Rule:
Recursive Rule:
U0 = 7
Un = (Un-1) + 3
Explicit Rule is a Linear Function
U0 = 7
Un = (Un-1)*3
Explicit Rule is an Exponential Function
y  a  bx
y  7  3x
y  a(b) x
Strategy:
Try subtracting terms that are next to each other
to see if you get the same value.
EXAMPLE
(a) Write a recursive sequence
7, 21, 63, …
y  7(3) x
Strategy:
Try dividing terms that are next to each other
to check if you get the same value.
PRACTICE
(b) Write the explicit rule
(c) Find the 10th term in the sequence: u0, u1, u2, u3, u4, u5, u6, u7, u8, u9 … which would be u9
#1 Sequence: 5, 9, 13, …
#1 Sequence: 4, 12, 20, …
Solution: The initial value is 5 (y-intercept = 5)
The rule is add 4, Linear with slope =4
Recursive Rule:
U0 = 5
Un = (Un-1) + 4
Explicit Rule is a Linear Function
y  a  bx
y  5  4x
Find 10th term: 5,9,13,17,21,25,29,33, 37, 41
Or, use x=9 in y=5+4x to get y=5+4(9) = 41
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#2 Sequence: 5, 20, 80, …
Solution: The initial value is 5 (y-intercept = 5)
The rule is multiply by 4,
Exponential with multiplier =4
Recursive Rule:
U0 = 5
Un = (Un-1)4
Explicit Rule is a Exponential Function
y  a(b) x
y  5(4) x
Find 10th term: 5, 20, 80, 320, 1280, 5120, 20480,
81920, 327680, 1310720
Or, use x=9:
y  5(4) x  5(4)9  1310720
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#2 Sequence: 5, 40, 320, …
Example
#3 Given the recursive function:
Practice
#3 Given the recursive function:
an  an1  7 with a0  5
a
a
Find the values 1 and 4
an  an1  8 with a0  24
a
a
Find the values 2 and 4
Solution:
Use the rule: add 7 to generate the first few
terms of this sequence.
Start with the initial value 5.
a0, a1, a2, a3, a4
5, 12, 19, 26, 33, …
a1  12
and
a4  33
#4 Given the recursive function:
#4 Given the recursive function:
1
an   an 1    with a0  500
2
a
a
Find the values 1 and 6
1
an   an 1    with a0  4096
4
a
a
Find the values 1 and 6
Solution:
Use the rule: multiply by one-half to generate the
first few terms of this sequence.
Start with the initial value 500.
a0, a1, a2,
a3,
a4,
a5, a6
500, 250, 125, 62.5, 31.25, 15.625, 7.8125, …
a1  250
and
a6  7.8125
#5 “Challenge” Given the recursive function:
1
an   an 1   
2
Find
#5 “Extra Credit Challenge Problem”
a0  500
with
Given the recursive function:
1
an   an 1    with a0  4096
4
a100
Find
Solution: Use the explicit rule with x=99
x
y  a(b) x
1
1
y  500    500  
2
2
28
y  7.88 x 10
99
a100