Fibonacci Multiples
The sequence \(1, 1, 2, 3, 5, \ldots\) is called the Fibonacci Sequence. It is the sequence \(f(1)\), \(f(2)\), \(f(3)\), \(f(4)\), \(\ldots\) defined by
- \(f(1) = 1\),
- \(f(2) = 1\), and
- \(f(n + 2) = f(n + 1) + f(n)\) for all \(n \ge 1\).
So for example \(f(12) = 144\).
| Part A | Determine the smallest positive integer \(a\) such that \(F_a\) is a multiple of \(100\). | |
|---|---|---|
| Part B | Determine the smallest positive integer \(b\) such that \(F_b\) is a multiple of \(103\). | |
| Part C | Determine the smallest positive integer \(c\) such that \(F_{c-1}\) is a multiple of \(100\) and \(F_{c+1}\) is a multiple of \(103\). |
Hexagonal Squares
The square numbers \((1, 4, 9, 16, 25, \ldots\) are the numbers which can be arranged to make a square array of dots. Analogously, the Hexagonal numbers \((1, 7, 19, 37, \ldots\) are the numbers which can be arranged to make a hexagonal array of dots. A "Hexagonal-Square" is any positive integer which is both a Hexagonal number and a Square number. So for example \(1\) is the first (smallest) Hexagonal-Square.
| Part A | What is the second Hexagonal-Square? | |
|---|---|---|
| Part B | What is the sum of the first \(6\) Hexagonal-Squares? | |
| Part C | What is the square root of the tenth Hexagonal-Square? |