1. 4.2A Arithmetic Explicit and Recursive Sequences
How to write recursive and explicit formulas?

2. Warm-Up
f(x)=3x – 2
f(x)=3x-2 x
f(x) 2 4 3 7 ?

3. Explicit Notation-Arithmetic
Essential Question:
How are arithmetic sequences written as an explicit formula?
Explicit Definition:
An explicit formula allows direct computation of any term for a sequence a1, a2, a3, , an,

4. Developing the Explicit Formula for an Arithmetic Sequence
1, 5, 9, 13, 17, 21,...
Explicit Formula
a1  =  1
            a2  =  1+4
a3   = 1+4+4
a4  =  
an   =
  =  1
             =  1+4(1)
  =  1+4(2)
 =  1+4(3)
= 1+4(n - 1)
d = common
difference 4 1 n

5. Example 1a: Write an Explicit Equation+3

6. Example 1b: Use the Explicit Equation

7. Write the explicit equation for the sequence
Example 2a
Write the explicit equation for the sequence
{18, 13, 8, 3,-2…} ? -5

8. Example 2b

9. Example 3
Describe the pattern. What is happening to each term to get the next? Write the explicit equation.
Complete the pattern 3, 6, 9, ___,___,___,...
Complete the pattern 5, 10, 15, __,__,__…
12, 15, 18,
The numbers in the sequence are multiples of 3 – Inceasing by 3 each time
an = 3n Explicit Equation
20, 25, 30,
The number in the sequence are multiples of 5 – Adding 5 each time.
an = 5n Explicit Formula

10. 4.2A Recursive Notation - Arithmetic
Essential Question:
How are arithmetic sequences written as a recursive formula?
Recursive Definition:
Recursive formula is a formula that is used to determine the next term of a sequence using one or more of the preceding terms. The recursive sequence can be denoted by f(n) or fn

11. Warm Up
Using the given information. Write the next 3 terms in the sequence.
The first term is 9, ___,___,___,...
The first term is 1, ___,___,___…
12, 15, 18,
The previous term is being increase by 3
4 , 16, 64,
The previous term is being multiplied by 4

12. Developing the Recursive Formula for an Arithmetic Sequence
1, 5, 9, 13, 17, 21,...
Recursive Formula
Recursive Formulas have two parts
The starting value of a1.
The recursive equation for an as a function of an-1 (previous term)
a1  =  1   
           a2  =  1+4
a3   = (1+4)+4
a4  =  (1+4+4)+4
a5   = ( )+4
an   =
 =  1
           =  1   + 4
 =  5   + 4
 =  9   + 4
 = 13  + 4
 = an-1  + 4
an   = an-1  + d
previous term
previous term
Common difference
Common difference

13. Why?
Why are we learning how to write two different formulas for the same sequence?????

14. Example 4 2, 4, 6, 8,... 2 a1 = _____ an = an-1 + d an = an-1 + 2
Find the recursive equation for the following arithmetic sequence.
2, 4, 6, 8,... 2
Remember
Recursive Formulas have two parts
The starting value of a1.
The recursive equation for an as a function of an-1 (previous term)
a1   = _____
an   = an-1  + d
an   = an-1  + 2

15. Example 5 5, 8, 11, 14, 17,... a1 = _____ 5 an = an-1 + d
Find the recursive equation for the following arithmetic sequence.
5, 8, 11, 14, 17,...
a1   = _____ 5
an   = an-1  + d
an   = an-1  + 3