11/21/2016
(66) Recursive formulas for arithmetic sequences | Constructing arithmetic sequences | Sequences | Algebra I | Khan Academy
Recursive formulas for arithmetic sequences
Learn how to find recursive formulas for arithmetic sequences. For example, find the
recursive formula of 3, 5, 7,...
Share
Tweet
Email
Before taking this lesson, make sure you are familiar with the basics of
arithmetic sequence formulas.
How recursive formulas work
Recursive formulas give us two pieces of information:
1. The first term of the sequence
2. The pattern rule to get any term from the term that comes before it
Here is a recursive formula of the sequence 3, 5, 7, ...along with the
interpretation for each part.
a(1) = 3


a(n) = a(n − 1) + 2
← the first term is 3
← add 2 to the previous term
th
In the formula, n is any term number and a(n) is the n term. This
th
means a(1) is the first term, and a(n − 1) is the term before the n term.
In order to find the fifth term, for example, we need to extend the sequence
term by term:
https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/a/writing-recursive-formulas-for-arithmetic-sequences
1/5
11/21/2016
(66) Recursive formulas for arithmetic sequences | Constructing arithmetic sequences | Sequences | Algebra I | Khan Academy
a(n) = a(n−1) + 2
a(1)
=3
a(2) = a(1) + 2
=3+2 =5
a(3) = a(2) + 2
=5+2 =7
a(4) = a(3) + 2
=7+2 =9
a(5) = a(4) + 2
= 9 + 2 = 11
Cool! This formula gives us the same sequence as described by 3, 5, 7, ...
Check your understanding
1) Find b(4) in the sequence given
b(1) = −5
by

b(n) = b(n − 1) + 9
b(4) = Check
[I need help!]
Writing recursive formulas
Suppose we wanted to write the recursive formula of the arithmetic
sequence 5, 8, 11, ...
The two parts of the formula should give the following information:
The first term (which is 5)
https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/a/writing-recursive-formulas-for-arithmetic-sequences
2/5
11/21/2016
(66) Recursive formulas for arithmetic sequences | Constructing arithmetic sequences | Sequences | Algebra I | Khan Academy
The rule to get any term from its previous term (which is "add 3")
Therefore, the recursive formula should look as follows:
c(1) = 5


c(n) = c(n − 1) + 3
Check your understanding
2) What is the recursive formula of the
sequence12, 7, 2, ... ?
d(1) = 12


d(n) = −5
d(1) = 12


d(n) = d(n − 1) + 5
d(1) = 12


d(n) = d(n − 1) − 5
d(1) = 12


d(n) = −5 ⋅ d(n − 1)
https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/a/writing-recursive-formulas-for-arithmetic-sequences
3/5
11/21/2016
(66) Recursive formulas for arithmetic sequences | Constructing arithmetic sequences | Sequences | Algebra I | Khan Academy
Check
[I need help!]
3) Complete the missing values in the recursive
formula of the sequence 2, 8, 14, ...
e(1) = A


e(n) = e(n − 1) + B
A = B = Check
[I need help!]
4) Complete the missing values in the recursive
formula of the sequence −1, −4, −7, ....
f (1) = A


f (n) = f (n − 1) + B
A = B = Check
[I need help!]
Reflection question
https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/a/writing-recursive-formulas-for-arithmetic-sequences
4/5
11/21/2016
(66) Recursive formulas for arithmetic sequences | Constructing arithmetic sequences | Sequences | Algebra I | Khan Academy
Reflection question
5) Here is the general recursive formula for arithmetic
sequences.
g(1) = A


g(n) = g(n − 1) + B
What is the common difference of the sequence?
A
B
A+B
B−A
Check
[I need help!]
https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/a/writing-recursive-formulas-for-arithmetic-sequences
5/5