1. recursive formulas
Sequences: Lesson 3

2. Recursive formula
A Recursive Formula is a formula that calculates each term of a sequence based on the preceding term (term that came right before).
A recursive formula always uses the preceding term to define the next term of the sequence. You have to know what came before it
Recursive formulas can be used for arithmetic or geometric sequences
Recursive formulas look a little different than explicit. They will have an an-1 somewhere in the equation which represents the term prior to the term you are solving for, and they will give you a specific value

3. Write recursive formulas for Arithmetic Sequences
Ex. an = an-1 + 7
Means arithmetic with d=7,
so take the prior term and just add 7 to find the next
Means take the previous term and add the d to continue the pattern
The rules for recursive formulas may look more complicated, but are actually very logical (and often easier to use than explicit).

4. Write recursive formulas for Arithmetic Sequences
EXAMPLE: Write the recursive formula for the sequence: 2, 4, 6, 8…
A recursive formula tells us how each term is connected to the next term
It is arithmetic and the difference between each term is 2. We can display this in a recursive formula using the following:
an = an where a1=2
an = term number and an-1 = the term before the n term

5. Write recursive formulas for geometric Sequences
Ex. an = (an-1)(-0.5)
Means geometric with r=-.5
so take the prior term and just multiply by -.5 to find the next
Means take the previous term and multiply by r to continue the pattern

6. Write recursive formulas for geometric Sequences
EXAMPLE: Write the recursive formula for the sequence: 3, 6, 12, 24…
A recursive formula tells us how each term is connected to the next term
It is geometric and the ratio between each term is 2. We can display this in a recursive formula using the following:
an = (an-1)(2) where a1=3
an = term number and an-1 = the term before the n term

7. Use recursive formulas for arithmetic sequences:
Find the 5th term: an = an-1 + 2
2, 4, 6, …
We know the 3rd term in this sequence is 6 and it is arithmetic with d=2.
So, logically, the 4th term is 8 and the 5th term is 10.
Here’s how to do that with the formula:
a3 = 6 plug in 6 as an-1 when finding the 4th term since it was the term before it
an = an-1 + 2
a4 = a(4-1) + 2
a4 = a3 + 2
a4 = 6 + 2
a4 = 8
do it again:
a4 = 8 plug in 8 as an-1 when finding the 5th term since it was a4
a5 = a4 + 2
a5 = 8 + 2
a5 = 10
You MUST know the term that came before the one you want

8. Use recursive formulas for arithmetic sequences:
an = an-1 – a1 = 27
Find the first 5 terms of the sequence
We are provided the 1st term of the sequence, 27. We need to find the next four terms.
a2 = 27 – 2
a2 = 25
a3 = 25 – 2
a3 = 23
a4 = 23 – 2
a4 = 21
a5 = 21 – 2
a5 = 19
The first five terms are 27, 25, 23, 21, and 19.
Remember an-1 really just means the previous term

9. Use recursive formulas for geometric sequences:
Find the 19th term an = 2.5(an-1)
a18 = 10
We know the 18th term in this sequence is 10 and it is geometric with r=2.5
So, logically, the 19th term is 25.
Here’s how to do that with the formula:
a19 = 2.5(a(19-1))
a19 = 2.5(a18)
a19 = 2.5(10)
a19 = 25

10. Use recursive formulas for geometric sequences:
Find the 27th term an = 3(an-1)
a24 = 4
a24 = 4
a25 = 3(a24)
a25 = 3(4)
a25 = 12
a26 = 3(a25)
a26 = 3(12)
a26 = 36
a27 = 3(a26)
a27 = 3(36)
a27 = 108

11. 9, 1, -7, -15… Write a recursive formula for the sequence.
You subtract 8 to get to the next term.
an = a(n-1) – 8, where a1 = 9
Which formula would you use to find the 38th term?
The explicit formula is best for finding specific terms in a sequence. an = -8n + 17
an = -8n + 17
a38 = -8(38) + 17
a38 =
a38 = - 287

12. STRETCH: Write the first 5 terms of the sequence using the explicit formula given. Then, write the recursive formula for the sequence.
an = 2n + 10
Substitute the term numbers 1 through 5 for “n” to write the first 5 terms of the sequence.
12, 14, 16, 18, 20
How would you write the recursive formula?
Each term is increased by 2. Just add two to the previous term.
an = a(n-1) + 2, where a1 = 12
Why do we have to say what a1 is?

13. recap