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 = a in words: 5th term equals the 4th term plus 3
a5 =
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. Simplify: an = ___ + ___n
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 = ( n - 1) which simplifies to: an = n
**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 = ( n - 1) which simplifies to: an = n
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 = n
66 = 3n
22 = n