Section 1.1 Recursively Defined Sequences
Patterns are all around you. They are present in nature, science, architecture, and math. In math,
recursion is used to represent these patterns. Recursion is a process in which each step of a pattern is
dependent on the step or steps that come before it.
Example 1
A square table seats 4 people. Two square tables pushed together seat 6 people. Three tables pushed
together seat 8 people. Draw a picture to represent each of these scenarios.
1 square table:
2 square tables:
3 square tables:
How many people can sit at 10 tables arranged in a straight line? How many tables are needed to seat 32
people? Write a recursive definition to find the number of people who can sit at any linear arrangement of
square tables. Use a table to help see the pattern.
Tables
1
2
3
4
5
6
7
8
9
10
People
Graphical representation of the data points 
Why are the points on the graph not connected?
Recursive Definition:
u1 =
un =
How many tables are needed to sit 32 people?
Definitions:




A _____________ is an ordered list of numbers. The tables above represent the sequence 4, 6, 8, …
Each number in the sequence is called a ___________. The first term, u 1 (pronounced ‘u sub one’), is
4. The second term, u2, is 6, and so on.
The nth term, un, is called the _____________ of the sequence.
A recursive formula, the formula that defines a sequence, must specify at least one starting term (u 1)
and a recursive rule that defines the nth term in relation to a previous term or terms (un).
Example 2
A concert hall has 59 seats in the first row, 63 seats in the second row, 67 seats in the third row, and so
on. The concert hall has 35 rows of seats. Write a recursive formula to find the number of seats in each row.
How many seats are in Row 4? Which row has 95 seats?
Once you have a recursive definition, you can use it to find the other terms of the sequence.
u1 =
un =
To find the other terms in the sequence:
u1 =
u2 = u1 + ___ = ___ + ___ = ___
u3 = u2 + ___ = ___ + ___ = ___
u4 =
u5 =
u6 =
How many seats are in Row 4?
Which row has 95 seats?
The recursive definition un = un – 1 + ___ , is called an arithmetic sequence because it has a common
difference. A common difference is a constant number that is being added or subtracted from the previous
term.
To check and see if a sequence is an arithmetic sequence take the second number and subtract the first. Then
take the third number and subtract the second, and so on. If each term in the sequence has a constant number
as their difference, it is an arithmetic sequence.
Example 3
1, 5, 9, 13, 17, …
Check:
5–1=4
9–5=4
13 – 9 = 4
17 – 13 = 4
Since the difference is constant for each term, this is an arithmetic sequence.
Example 4
1, 3, 9, 27, …
Check:
3–1=2
9–3=6
27 – 9 = 18
If there is not a common difference, it may not be an arithmetic sequence, but a geometric sequence.
A geometric sequence is a sequence of numbers that is being multiplied by a constant, which is called the
common ratio. To check to see if a sequence is a geometric sequence, take the second number and divide it
by the first number. Then take the third number and divide it by the second, and so on. If each quotient is
constant, it is a geometric sequence and that constant quotient is your common ratio.
1, 3, 9, 27, …
Check:
3/1 = 3
9/3 = 3
27/9 = 3
So this is a geometric sequence with a common ratio of 3.
The recursive definition for a geometric sequence is in the form un = r ∙ un – 1. So the above sequence would be
un = 3 ∙ un – 1 and the u1 = 1.
Example 5
The geometric pattern below is created recursively. If you continue the pattern endlessly, you create
a fractal called the Sierpinski triangle. A fractal is a geometric result of infinitely many applications of a
recursive procedure or calculation. How many black triangles are there at Stage 20? Use your calculator notes
to help you.