# Lesson 3: Recursion Formulas

```Unit 7
MCR 3U1
Lesson 3: Recursion Formulas
Recall: A formula for calculating the nth term of a sequence without knowing any previous terms
is called an explicit formula.
Using the diagram to the right,
we observe the following:
 Diagram 1 is a square with a
side length of 1
 Diagram 2 was drawn by using
Diagram 1
Diagram 2
Diagram 3
Diagram 1, and then another
square was drawn adjacent to
it to create a rectangle
 Diagram 3 was drawn by using Diagram 2 and drawing a square with side length 2 units
adjacent to and directly above the two smaller squares.
Answer the questions that follow given the diagram and the information above.
1. Diagram will contain a square with side length 3 units. Where should
this square be drawn to continue this pattern? Draw Diagram 4.
2. Complete the following table for the rest of this pattern.
Diagram Number
1
2
3
4
5
6
Side Length of Square
1
1
2
3. Write the side lengths of the squares as a sequence.
This pattern is a special type of sequence known as the Fibonacci sequence. This famous
sequence consists of the numbers 1, 1, 2, 3, 5, 8, 13, …. Each number, after the first two
numbers, is the sum of the two preceding terms.
How does this help us? Well, we can now work with a formula that uses terms within a sequence
to create new terms. A formula used to calculate a term in a sequence from one or more
previous terms is called a recursion formula.
Unit 7
MCR 3U1
A recursion formula consists of at least two points. The first part(s) give the values of the
first term(s) in the sequence. The last part is an equation that can be used to calculate each of
the other terms from the term(s) before it.
Example 1: Write the first 5 terms of the sequence determined by the recursion
formula
t1  11
t n  t n 1  4
. Determine the explicit formula for this sequence.
Example 2: Write the first 5 terms of the sequence determined by the recursion
formula
t1  2
t n  3t n1
. Determine the explicit formula for this sequence.
```