Recursive Sequences

In order to precisely define a recursive sequence, we need two parts: the relationship relating the next term to previous terms, and as many original terms as are needed to plug into the recursive formula.

As an added bonus, compute:

1. $a_0$
2. $a_{-2}$
3. $a_6$

A related problem is the problem of guessing the function based off a few terms and other information. There is no way to generally solve this (it is easy to produce infinite possibilities), but one can make some educated guesses. In fact, this is essential when trying to find patterns while problem solving.

Think about this sequence: $$1,1,2,3,5,8,\dots$$ This is the famous Fibonacci Sequence. This sequence is a perfect example of a recursive sequence. A recursive sequence is defined off of preceding terms. This sequence has the pattern for the $n$th term $t_n$, $t_n=t_{n-1}+t_{n-2}$.

As we see, recursive sequences can include things like arithmetic sequences and geometric sequences.

Terms in a recursive sequence have closed forms. For example, the closed form of $1,2,4,8..$ is $t_n=2^n$. Let's try a problem:

Good job! Hopefully you learned something new about recursion today.

Write out the terms of the recursive sequence for $a_n=a_{n-1}^2-3$ for $a_1=1$ from $1$ to $6$. You might notice a pattern!

If $a_n=a_{n-k}$ for all $n$, and $k$ is the smallest such integer, $k$ is known as the period of the recursion. If $k=3$, and the recursion only depends on the three previous terms, it will always repeat. Think about why this might be true. This can be developed into a form of mathematical proof known as mathematical induction, which depends on another principle called the well-ordering principle.

By the way, the first six terms are $1, -2, 1, -2, 1,$ and $-2$. Try to show that this will go on forever in this pattern.

Recursive Sequences

Our first lesson in sequences will be about recursive sequences!

A recursive sequence is a type of sequence where the next term depends on the previous term. Here is an example a of recursive sequence:

$$a_5 = a_4 + 7$$

This sequence is recursive because we cannot calculate the value of $a_5$ without knowing the value of $a_4$.

(Video) Recursive Sequences

by PatrickJMT

In this video by PatrickJMT, you will walk through an example of recursive sequences.

Summary

This video explains how to find the first few terms of a sequence using a formula. It shows how to use a given first term to find the next terms one at a time.

The first example finds the first three terms by plugging each value of $n$ into the formula. The video then works through a second example where the formula includes fractions and negative signs. You will see how to use the previous term to find each new term in the sequence.

The video also shows how the terms can form a pattern, such as numbers that increase while their signs alternate. By the end, you will understand how to substitute the correct value of $n$ and use the formula to find the terms of a sequence.

Recursive Sequences

A recursive sequence is a list of numbers where each number depends on one (or more) number that comes before it. Instead of giving a formula to jump at any term, it tells you how to build the sequence step-by-step.

To write a recursive sequence, you need:

1. A starting value (or the first term)
2. A rule that tells you how to get the next term

Example

Let's say that this is a recursive rule:

$$a_1 = 2, a_n=a_{n-1} +3$$

This means that the first term is $2$, and each term after that is $3$ more than the one before. So the sequence is: $$2, 5, 8, 11, 14, 17, ...$$

You can only find the next term if you know the one before it. That's what makes this sequence recursive.

Special Types

Some recursive sequences use more than one previous term. A famous one is the Fibonacci sequence: $$F_1 = 1, F_2=1, F_n=F_{n-1}+F_{n-2}$$

This means that each term is the sum of the two before it: $$1,1,2,3,5,8,13,...$$

Recursive sequences are used in computer science, biology, finance, and more. They're helpful when you want to build step-by-step instead of jumping straight to a formula.