Recursive and Explicit Forms of Arithmetic Sequences

1
Recursive and Explicit Formulas for Arithmetic
(Linear) Sequences
2
An arithmetic sequence is a sequence with a
constant increase or decrease also known as the
constant difference In the sequence 10, 40, 70,
100, . The constant difference between the
terms is 30
3
A recursive formula for a sequence would be
a1 first term in the sequence an term you are
trying to find an-1 previous term in the
sequence d constant difference
4
In the sequence 10, 40, 70, 100, . The
constant difference between the terms is 30 The
first term of the sequence is 10 The recursive
formula for this sequence would be
I plugged in 10 for the first term and 30 for
the constant difference
Write the recursive formula of the sequence 4, 7,
10, 13, .
5
In the sequence 4, 7, 10, 13, . To find the
5th term recursively, I plug it into the formula
I just made an an-1 3 a5 a5-1 3
in words 5th term equals the 4th term plus
3 a5 13 3 a5 16
6
Explicit Formulas

• A formula that allows you to find the nth term of
  the sequence by substituting known values in the
  expression.

An explicit formula for sequence would be
an a1 d( n - 1)
a1 first term in the sequence an current term
in the sequence d constant difference n term
number
7

• In the sequence 10, 40, 70, 100, .
• The constant difference between the terms is 30
• The first term of the sequence is 10

The explicit formula for this sequence would be
an 10 30( n - 1) which simplifies to an
-20 30n
I plugged in 10 for the first term and 30 for
the constant difference and distributed
Write the explicit formula of the sequence 4, 7,
10, 13, .
an ___ ___( n - 1) Simplify an ___
___n
8
In the sequence 4, 7, 10, 13, . To find the
11th term explicitly, I plug in the into the
formula I just made an 1 3n a11 1
3(11) a11 34
Find the 15th term of the sequence using the
formula an 1 3n
9
Example Test Question
The first row of the theater has 15 seats in it.
Each subsequent row has 3 more seats that the
previous row. If the last row has 78 seats, how
many rows are in the theater?
Step 1 Create an explicit formula
an 15 3( n - 1) which simplifies to an
12 3n
Step 2 Figure out what you are solving for the
last row number tell you the number of rows in
the theater and you know the last row (an) has 78
seats
Step 3 Solve for the last row number 78 12
3n 66 3n 22 n