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## How can recursion and induction be used to describe and analyse situations?

### Key questions

1. 1

What are the Fibonacci numbers, and what properties do they have?

2. 2

How can the highest common factor of two numbers be found?

3. 3

What is the Fundamental theorem of arithmetic, and why does it matter?

4. 4

What does it mean to describe situations recursively?

5. 5

How can results be proved using induction?

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#### Introducing...

Resource type Title
Investigation Picture this!

#### Developing...

Resource type Title
Rich example Euclid's algorithm
Building blocks One step, two step
Building blocks Triominoes
Scaffolded task The Fundamental Theorem of Arithmetic
Problem requiring decisions Multiplication magic square
Food for thought A Diophantine equation
Food for thought Factorial fun
Food for thought LCM Sudoku
Investigation Buckets and ponds
Go and think about it... $S$-prime numbers

#### Review questions

Title Ref
Are any two Fermat numbers relatively prime? R8677
Can we find a solution to $(n-3)^3+n^3=(n+3)^3$? R9919
Can we find integers that satisfy $a^3 + 3b^3 = 9c^3$? R7096
Can we prove these Fibonacci number results? R9868
Can we show the sum of this series to $n$ terms is $n/(3n-1)$? R5634
Can we write 33127 as the difference of two squares? R6171
For how many integers $n$ is $n/(100-n)$ also an integer? R8880
If $x^3 = 2x+1$, what is $x^k$ as a quadratic? R8586
Two rules generate a sequence; what do the first 100 terms add to? R7943
What is the primorial of a number? R8811
What's the next number to occur in both sequences? R9916
What's the smallest integer with 426 proper factors? R7465