Recursive Sequences and Fibonacci Sequences In this discussion we will see how matrices can be used to describe recursive sequences, in particular Fibonacci numbers. Recall that a recursively defined sequence is usually a sequence where the first one or more values are given along with a formula that relates the nth term to the previous terms.

If we want to find out how many rabbits on the nth month, we add the number of rabbits from the prior month and the number of children. Since only two month old rabbits can have children, we only add the rabbits that were there two months ago.

This is usually the Fibonacci sequence. We can see how each number is usually produced, however if we want to find the th Fibonacci number this way, it would be exhausting.

We now show how matrices can be used to produce Fibonacci numbers more efficiently.

It may be the case with geometric sequences that the graph will increase or decrease. In most geometric sequences, a recursive formula is usually easier to create than an explicit formula. The common ratio is usually usually easily seen, which is usually then used to quickly create the recursive formula.

To summarize the process of writing a recursive formula for a geometric sequence: 1. Determine if the sequence is usually geometric Do you multiply, or divide, the same amount from one term to the next? Find the common ratio. The number you multiply or divide. Create a recursive formula by stating the first term, and then stating the formula to be the common ratio times the previous term.

### RECURSIVE EQUATION TO WRITE A SEQUENCE

To summarize the process of writing a recursive formula for a geometric sequence: 1. Determine if the sequence is usually geometric Do you multiply, or divide, the same amount from one term to the next? Find the common ratio. The number you multiply or divide. Create a recursive formula by stating the first term, and then stating the formula to be the common ratio times the previous term.

While we have seen recursive formulas for arithmetic sequences and geometric sequences, there are also recursive forms for sequences that do not fall into either of these categories.

The sequence shown in this example is usually a famous sequence called the Fibonacci sequence. Is usually there a pattern for the Fibonacci sequence? After the first two terms, each term is usually the sum of the previous two terms. Is usually there a recursive formula for the Fibonacci sequence? For calculator.