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SAT Subject Test Mathematics Level 2: Practice and Study Guide25 chapters | 240 lessons | 15 flashcard sets

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After reading this lesson, you'll know how to identify recursive functions and how they work. You'll learn how to figure out the terms of these recursive functions, and you'll learn about a famous recursive function.

A **recursive function** is a function that calls itself, meaning it uses its own previous terms in calculating subsequent terms. This is the technical definition. Now, let's look at what this means in a real-world math problem.

This is a real-world math recursive function. This is actually a really famous recursive sequence that can be seen in nature.

What makes this function a recursive one is that it requires its own terms to figure out its next term. For the function we are looking at right now, to find the nth term, you need to know the previous term and the term before that one. So, you need to know the previous two terms for this special sequence.

This is how you can determine whether a particular function is recursive or not. If the function requires a previous term in the same sequence, then it is recursive.

Note how this function specifically states the beginning two values. Most recursive functions will give you the beginning value or values that are needed to fully calculate the sequence. Without these beginning values, there is no way to determine what the real values for each term should be.

Now, let's look at how you work with this recursive formula.

Recursive functions are usually sequences. It gives you the formula to find the next term, if you know the previous terms. So, to calculate your terms from a recursive formula, you begin by writing out your beginning numbers. So with the beginning recursive formula that you saw, the first two terms in the sequence are 1 and 1. Now, to find the third number in the sequence, when n is 3, you calculate your function for (a sub 3).

This function is telling you that the third term in the sequence is equal to the second term plus the first term. So, the third term is equal to 1 + 1 = 2.

The fourth term, according to the function, is this:

You begin to see the pattern here. Each term is the sum of the previous two terms.

n | Value |
---|---|

1 | 1 |

2 | 1 |

3 | 1 + 1 = 2 |

4 | 2 + 1 = 3 |

5 | 3 + 2 = 5 |

6 | 5 + 3 = 8 |

And there you have the beginning of this famous recursive function: 1, 1, 2, 3, 5, 8.

This particular recursive function gives you the famous Fibonacci sequence. This sequence is found in nature. If you count the number of petals of most flowers, you'll see that the number will almost always be one of the numbers in this sequence. You'll find flowers with 5 petals, 13 petals, or 21 petals. All are numbers in the Fibonacci sequence. You can check these numbers by continuing to calculate out the sequence.

Let's look at another example.

Calculate the first 5 terms of this recursive function:

To find the first 5 terms of this recursive function, you begin by writing down the given term, in this case, 1. Then, you use this to calculate your next terms. Looking at this function, you see that the next term is the previous term plus 2.

Continuing this pattern, you calculate your first 5 terms to be these:

n | Value |
---|---|

1 | 1 |

2 | 1 + 2 = 3 |

3 | 3 + 2 = 5 |

4 | 5 + 2 = 7 |

5 | 7 + 2 = 9 |

Your first 5 terms are 1, 3, 5, 7, 9. Look at that, this sequence happens to be the sequence of odd numbers.

Let's review. A **recursive function** is a function that use its previous terms to figure out its next term. You are usually given the necessary beginning terms. And then, your job will be to calculate the next terms by following the function and using the previous terms as needed. A really famous recursive function is the Fibonacci sequence that can be seen in flower petals.

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SAT Subject Test Mathematics Level 2: Practice and Study Guide25 chapters | 240 lessons | 15 flashcard sets

- Go to Exponents

- Go to Vectors

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- Parametric Equations in Applied Contexts 5:29
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- Go to Matrices

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