Aim #86: How do we write a recursive formula for a sequence?
Homework: Handout
Do Now: What is the next term in the sequence 5, 8, 11, 14, 17, ...
When asked to find a formula for this sequence, Ashley wrote the following on a
piece of paper:
Use Ashley's reasoning to help create a formula:
an =
What does a n represent?
Ashley's formula is called an explicit formula. You can use the formula to find the
value of any term you want without having to know the value of the term before it.
For example, if you wanted to know the 88th term, just substitute 88 for n and
evaluate.
When Johnny saw the sequence, he wrote the following:
an+1 = an + 3 and a 1 = 5.
What do we call the 5
th
term?
th
How could we find the 5 term in terms of the 4
If we want the 6
th
term in terms of the 5
If we want the (n + 1)
th
th
th
term?
term?
term in terms of the n
th
term?
The statement an+1 = an + 3, a1 = 5 is a recursive formula. A recursive formula
relates a term in the sequence to the preceding term or terms of the sequence.
Would it be equivalent to write the sequence as an = an-1 + 3? Why or why not?
If we wanted the 200th term of the sequence, which formula would be more
useful?
If we wanted to know how the sequence changes from one term to the next, which
formula would be more useful?
Using Johnny‛s recursive formula, what would we need to know if we wanted to
find the 200th term?
Write the first 5 terms of the following recursive sequences:
a. an+1 = 3an where a1 = -2
b. f(n) = f(n-1) - 8 where f(1) = 20
c. an = .25an-1 + 4 where a1 = 16
d. f(n+1) = -2f(n) + 1 where f(1) = -5
Write the recursive formula for the following sequences:
a. 2, 5, 8, 11, 14
b. -2, 1, -0.5, 0.25, -0.125
c. 3, 10, 31, 94, 283
d. 1 - x, 3, 5 + x, 7 + 2x, 9 + 3x
Consider a sequence given by the formula a n = a n-1 - 5, where a1 = 12.
a. List the first five terms of the sequence.
b. Write an EXPLICIT formula for the sequence.
c. Find the value of a 6 .
d. Find the value of a 100 .
For each sequence below, an explicit formula is given. Write the first 5 terms
of each sequence. Then, write a recursive formula for the sequence.
a. an = 2n + 10 for n ≥ 1
b. an =
1 n-1
for n ≥ 1
( ) 2
Let's sum it up!!!
An explicit formula specifies the nth term of a sequence as an expression in n.
(You can jump straight to the term you want)
A recursive formula specifies the nth term of a sequence as an expression in
the previous term (or previous couple of terms).