Sequences - Fibonacci

Consider the following assumptions about a rabbit population.

• The population starts with a single newly born pair (one male and one female).
• Each pair of rabbits produces a new pair at monthly intervals.
• Each pair of rabbits produces its first offspring two months after the pair was born.

This problem was addressed by Leonardo of Pisa (or Fibonacci) in 1202 in his book Liber Abaci.  The numbers are those of the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

For material and problems involving the Fibonacci sequence, go to Ron Knott's site.

A person wants to make a path 10 metres long by 1 metre wide out of concrete pavers.

Each paver is a 1 metre by ½ metre rectangle.

How many different ways can the pavers be laid without cutting any of them?

(two solutions - one involving the use of combinations)

Use mathematical induction to prove Binet’s formula for the n-th term of the Fibonacci sequence.

(Assume the formula is true for n = k and n = k + 1 and show that the formula is true for n = k + 2.)