Readers should be wary: some authors give the Fibonacci sequence with the initial conditions (or equivalently ). This change in indexing does not affect the actual numbers in the sequence, but it does change which member of the sequence is referred to by the symbol and so also changes the appearance of certain identities involving the Fibonacci numbers.
As with many linear recursions, we can run the Fibonacci sequence backwards by solving its recursion relation for the term of smallest index, in this case . This allows us to compute, for example, that , , , and so on. Because these preceding terms are uniquely defined by the recursion, one frequently sees the definition of the Fibonacci sequence given in the form , and for . In general, one can show that .
and Binet's Formula
- Main article: Binet's formula
The ratios , , , , , ..., between successive terms of the sequence tend towards the limit , a constant often denoted (the Greek letter phi, also written ). is a solution of the quadratic equation . The other root is . One possible explanation for this fact is that the Fibonacci numbers are given explicitly by Binet's formula. It is . (Note that this formula is valid for all integers .) It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre.
The most important identity regarding the Fibonacci sequence is its recursive definition, . The following identities involving the Fibonacci numbers can be proved by induction.
- The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
- A colony has rabbit. A rabbit produces one offspring every month. An offspring rabbit takes one month to grow up. Find a formula for the number of rabbits (including offspring) in the th month.
- How about if the colony starts with rabbits and offspring?
- Use this result to prove the identity .
- Find .
- Prove the above identites.
- Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. (Manhattan Mathematical Olympiad 2004)
- Except for the first two terms, each term of the sequence is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer produces a sequence of maximum length?
- A fair coin is to be tossed times. Let , in lowest terms, be the probability that heads never occur on consecutive tosses. Find .
- Find if and are integers such that is a factor of .
- Determine the maximum value of , where and are integers satisfying and .