# How to write a sequence in math

Sequences can be studied simply as patterns, or for patterns, such as in the study of prime numbers.

And then each successive term, for a sub 2 and greater-- so I could say a sub n is equal to a sub n minus 1 plus 3.

The Fibonacci numbers are formed from the integer sequence whose elements are formed by taking the sum of the previous two elements.

### Sequence and series

Sequences can be studied simply as patterns, or for patterns, such as in the study of prime numbers. In fact, every real number can be written as the limit of a sequence of rational numbers, e. In cases such as this, there are other notations that can be both more natural and more useful. You get to 7. So for the second term, we add 2 once. Let's say we wanted to define it recursively. Be careful here.

This famous sequence is known as Recaman's sequence. If I wanted to write it recursively, I could say a sub 1 is equal to negative 5.

Listing is most useful for infinite sequences with a pattern that can be easily discerned from the first few elements. Now, how does this make sense?

Important examples[ edit ] A tiling with squares whose sides are successive Fibonacci numbers in length There are many important integer sequences.

## Sequence examples english

Sometimes you may encounter a problem in an arithmetic sequence that involves fractions. Recursive and explicit sequences Video transcript What I want to do in this video is familiarize ourselves with the notion of a sequence. Well, once again, we could write this as a sub k. So in general, if you wanted a generalizable way to spot or define an arithmetic sequence, you could say an arithmetic sequence is going to be of the form a sub n-- if we're talking about an infinite one-- from n equals 1 to infinity. Other important examples of sequences include ones made up of rational numbers, real numbers, and complex numbers. So this, first of all, this is not arithmetic. So we could say, this is equal to a sub n, where n is starting at 1 and it's going to infinity, with-- we'll say our base case-- a sub 1 is equal to 1. That is how the terms in the sequence are generated. This sequence does not have any pattern that is easily discernible by eye, unlike the preceding sequence, which is increasing. But this is all they refer to. And here, we're going from our first term-- so this is a sub 1, this is a sub all the way to infinity. Illustrative Examples of Increasing and Decreasing Arithmetic Sequences Here are two examples of arithmetic sequences. So just to be clear, this is one, and this is one right over here.

Now we are adding 4. And let's say I only have these four terms right over here.

It is a good practice to write all the terms in the sequence and label them, if possible. So this right over here is explicit.

### Writing rules for sequences

And so then we could write a sub k is equal to the previous term. We add 7. For the second term, we added 4 once. We can obtain the next three terms by adding the last term by this common difference. We're going to add positive 2 one less than the index that we're looking at-- n minus 1 times. Now, there's a bunch of different notations that seem fancy for denoting sequences. Well, we're defining what a sub 1 is. So we go to 3, to 7, to 11, And so a sub 1 is going to be 1.

Starting at k, the first term, going to infinity with-- our first term, a sub 1, is going to be 3, now. So this is an explicit definition of it, but we could also do it recursively.

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