## Introduction to Sequences

A mathematical sequence is an ordered list of objects, often numbers. Sometimes the numbers in a sequence are defined in terms of a previous number in the list.

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### Key Takeaways

Key PointsThe number of ordered elements (possibly infinite ) is called the length of the sequence. Unlike a set, order matters, and a particular term can appear multiple times at different positions in the sequence.An arithmetic sequence is one in which a term is obtained by adding a constant to a previous term of a sequence. So the **sequence**: An ordered list of elements, possibly infinite in length.**finite**: Limited, constrained by bounds.**set**: A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects that may be contained within it.

### Sequences

In mathematics, a sequence is an ordered list of objects. Like a set, it contains members (also called elements or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and a particular term can appear multiple times at different positions in the sequence.

For example,

### Examples and Notation

### Finite and Infinite Sequences

A more formal definition of a finite sequence with terms in a set

ight }

ight }

ightarrow 2, 2

ightarrow 3, 3

ightarrow 5, 4

ightarrow 7, 5

ightarrow 11, cdots

A sequence of a finite length n is also called an

### Recursive Sequences

Many of the sequences you will encounter in a mathematics course are produced by a formula, where some operation(s) is performed on the previous member of the sequence

### Arithmetic Sequences

An arithmetic (or linear) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term. An example is *common * *difference* (

Another example is

In both of these examples,

### Geometric Sequences

A geometric sequence is a list in which each number is generated by multiplying a constant by the previous number. An example is *common ratio* (

Another example is

In both examples

### Explicit Definitions

An explicit definition of an arithmetic sequence is one in which the

To find the explicit definition of an arithmetic sequence, you begin writing out the terms. Assume our sequence is

and so on. From this you can see the generalization that:

which is the explicit definition we were looking for.

The explicit definition of a geometric sequence is obtained in a similar way. The first term is

## The General Term of a Sequence

Given terms in a sequence, it is often possible to find a formula for the general term of the sequence, if the formula is a polynomial.

### Key Takeaways

Key PointsGiven terms in a sequence generated by a polynomial, there is a method to determine the formula for the polynomial.By hand, one can take the differences between each term, then the differences between the differences in terms, etc. If the differences eventually become constant, then the sequence is generated by a polynomial formula.Once a constant difference is achieved, one can solve equations to generate the formula for the polynomial.Key Terms**sequence**: A set of things next to each other in a set order; a series**general term**: A mathematical expression containing variables and constants that, when substituting integer values for each variable, produces a valid term in a sequence.

Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence. Such a formula will produce the

If a sequence is generated by a polynomial, this fact can be detected by noticing whether the computed differences eventually become constant.

### Linear Polynomials

Consider the sequence:

The difference between

Suppose the formula for the sequence is given by

The difference between each term and the term after it is

So, the

### Quadratic Polynomials

Suppose the

This sequence was created by plugging in

If we start at the second term, and subtract the previous term from each term in the sequence, we can get a new sequence made up of the differences between terms. The first sequence of differences would be:

Now, we take the differences between terms in the new sequence. The second sequence of differences is:

The computed differences have converged to a constant after the second sequence of differences. This means that it was a second-order (quadratic) sequence. Working backward from this, we could find the general term for any quadratic sequence.

### Example

Consider the sequence:

The difference between

This list is still not constant. However, finding the difference between terms once more, we get:

This fact tells us that there is a polynomial formula describing our sequence. Since we had to do differences twice, it is a second-degree (quadratic) polynomial.

We can find the formula by realizing that the constant term is

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Next we note that the first item in our first list of differences is

Finally, note that the first term in the sequence is

So,

### General Polynomial Sequences

This method of finding differences can be extended to find the general term of a polynomial sequence of any order. For higher orders, it will take more rounds of taking differences for the differences to become constant, and more back-substitution will be necessary in order to solve for the general term.

### General Terms of Non-Polynomial Sequences

Some sequences are generated by a general term which is not a polynomial. For example, the geometric sequence

General terms of non-polynomial sequences can be found by observation, as above, or by other means which are beyond our scope for now. Given any general term, the sequence can be generated by plugging in successive values of

## Series and Sigma Notation

Sigma notation, denoted by the uppercase Greek letter sigma

ight ),

### Key Takeaways

Key PointsA series is a summation performed on a list of numbers. Each term is added to the next, resulting in a sum of all terms.Sigma notation is used to represent the summation of a series. In this form, the capital Greek letter sigma

ight )**summation**: A series of items to be summed or added.**sigma**: The symbol

Summation is the operation of adding a sequence of numbers, resulting in a sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. For finite sequences of such elements, summation always produces a well-defined sum.

A series is a list of numbers—like a sequence—but instead of just listing them, the plus signs indicate that they should be added up.

For example,

### Sigma Notation

One way to compactly represent a series is with *sigma notation*, or *summation notation*, which looks like this:

The main symbol seen is the uppercase Greek letter sigma. It indicates a series. To “unpack” this notation,

More generally, sigma notation can be defined as:

In this formula, *i* represents the index of summation, *.*

Another example is:

This series sums to

### Other Forms of Sigma Notation

Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context. For example:

## Recursive Definitions

A recursive definition of a function defines its values for some inputs in terms of the values of the same function for other inputs.

### Key Takeaways

Key PointsIn mathematical logic and computer science, a recursive definition, or inductive definition, is used to define an object in terms of itself.The recursive definition for an arithmetic sequence is:

### Recursion

In mathematical logic and computer science, a recursive definition, or inductive definition, is used to define an object in terms of itself. A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other inputs.

For example, the factorial function

This definition is valid because, for all

For example, we can compute

Putting this all together we get:

### Recursive Formulas for Sequences

When discussing arithmetic sequences, you may have noticed that the difference between two consecutive terms in the sequence could be written in a general way:

The above equation is an example of a recursive equation since the

In this equation, one can directly calculate the nth-term of the arithmetic sequence without knowing the previous terms. Depending on how the sequence is being used, either the recursive definition or the non-recursive one might be more useful.

A recursive geometric sequence follows the formula:

An applied example of a geometric sequence involves the spread of the flu virus. Suppose each infected person will infect two more people, such that the terms follow a geometric sequence.

**The flu virus is a geometric sequence:** Each person infects two more people with the flu virus, making the number of recently-infected people the nth term in a geometric sequence.

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Using this equation, the recursive equation for this geometric sequence is:

Recursive equations are extremely powerful. One can work out every term in the series just by knowing previous terms. As can be seen from the examples above, working out and using the previous term