Recursive Formulas and Explicit Formulas
Name: ____________________________ A#__
1. Consider the sequence 3, 8, 13, 18, 23, …
a) Write the recursive formula for this sequence.
b) Use the recursive formula to find the 7th term. Show your work.
c) How could you find the 25th term without using the recursive formula? Find the 25th term.
d) The 25th term of this sequence is 123. Was your answer to part c correct? If not, how could you adjust your strategy
in part c to get the correct answer?
2. What is an explicit formula of a sequence? How is it different than a recursive formula?
3. All of the sequences we have looked at the past two days have been arithmetic sequences.
a) Describe how arithmetic sequences change.
b) What vocab word did we use to describe how the sequence changed? What letter did we use to represent that term?
c) What do their graphs look like?
4. There are two explicit formulas for arithmetic sequences. We will use the sequence from problem #1 to see their
connection.
𝐴𝑛 = 𝐴1 + 𝑑(𝑛 − 1)
𝐴𝑛 = 𝐴0 + 𝑑𝑛
a) For each explicit formula write
out in words what each piece of the
equation means.
b) Fill in the unknowns from
problem #1 to write the explicit
formula for the sequence in each
form.
5. Using the explicit formula in the left hand column. Distribute and simplify the expression on the right hand side of
the equation.
6. How does this compare to the explicit formula in the right hand column.
7. Explain what is happening when you distribute and combine like terms that connect the two formulas.
8. We have learned that all linear functions are of the form 𝑦 = 𝑚𝑥 + 𝑏. Which of the two explicit formulas looks most
like our rule for linear functions? Explain the connection between each variable.
9. What’s the difference in an explicit formula for an arithmetic sequence and a linear equation?
Explicit Formula for Arithmetic Sequence
Linear Equation
What does the
variable represent?
What types of
numbers can you use
for each variable?
How does the graph
look?
For each arithmetic sequence below write the explicit and recursive formulas. Simplify each explicit formula completely.
10) 8, 17, 26, 35, 44, …
11) -83, 16, 115, 214, 313, …
Recursive Formula
Explicit Formula
5
12) 81, 74, 67, 60, 53, …
Recursive Formula
Explicit Formula
7
13) 2, 2, 3, 2, 4, …
Explicit Formula
14) 53, 41, 29, 17, …
Recursive Formula
Recursive Formula
Recursive Formula
Explicit Formula
15) 6.25, 7.5, 8.75, 10, …
Explicit Formula
Recursive Formula
Explicit Formula
16. Lucas is running a similar workout where he begins by running 100 meters, then 200 meters, then 300 meters, then
400 meters, and then finally 500 meters.
a) Is Lucas’ workout an arithmetic sequence or a geometric
sequence? Explain why.
b) Write a recursive and explicit formula representing
Lucas’ workout.
c) Use the explicit formula you found in part (b) to predict how many intervals it would take for Lucas to run a mile in
one interval (mile is about 1600 meters).