Sequences and Series

Note that Limits of Sequences are discussed here in the Limits and Continuity section.

Introduction to Sequences and Series

Sequences are basically just numbers or expressions in a row that make up some sort of a pattern; for example, January, February, March, …, December is a sequence that represents the months of a year. Each of these numbers or expressions are called a term or an elements of the sequence.

Sequences are the list of these items, separated by commas, and series are the sum of the terms of a sequence (if that sum makes sense; it wouldn’t make sense for months of the year).

You may have heard the term inductive reasoning, which is reasoning based on patterns, say from a sequence (as opposed to deductive reasoning, which is reasoning from rules or definitions). For example, try to find the next few terms in the following sequences:


We will do more like this in the Writing Formulas section below.

Summary of Formulas for Sequences and Series

For reference, here is a summary of the main formulas for Sequences and Series:

Sequences and Series Definitions

I like to compare sequences to relations or functions we learned about in the Algebraic Functions section. Think of the $ x$-part of the relation (the independent variable) as the numbers in a row that represent each part of the sequence; these typically start at 1 (one). Thus, the $ x$ part will typically be 1, 2, 3, and so on (natural numbers). This is usually represented by the variable $ n$.

The $ y$-part of the sequence is the actual term, or what the element is for that particular $ n$-value; this is usually represented by $ {{a}_{n}}$ (but sometimes you see this as $ f\left( n \right)$ instead). This can be any number.

Think of the “points” or “coordinates” of a sequence as $ \left( {n,{{a}_{n}}} \right)$.

For example, the sequence 3, 7, 11, 15, …, we have:


Note that sequences are said to be convergent if as the domain (the $ n$’s) approach infinity, the range (the $ {{a}_{n}}$’s) approach some number, called a limit. (We saw this when studying the end behavior of a function in the Parent Function and Transformations, and Graphing Rational Functions, including Asymptotes sections). A sequence is divergent if it is not convergent. We will talk about Limits of Sequences here in the Limits and Continuity section.

Explicit versus Recursive Formulas

Explicit formulas are formulas that are computed for each term; in other words, you can look at the formula for a term and know exactly how to get that term (you don’t rely on the previous term). We’ll talk about the two main types of explicit formulas, Arithmetic and Geometric, later.

Recursive formulas are formulas that use the previous term to get to the next one, and we always have to indicate what the first term is ($ {{a}_{{1}}}$), so we can get started. Notice that to reference the previous term, we use $ {{a}_{{n-1}}}$ (sometimes $ f\left( {n-1} \right)$), which makes sense, since we need the term before this one. Since the use of recursive formulas are very limited, we typically don’t deal with them very much.

Here are examples; for most of these, we’ll come up with some tools to make finding the formulas much easier. And don’t worry if you don’t see how to get these right away; these take practice!

Another Recursive Example

Here’s an example of a more complicated recursive formula; Get the first 5 terms: $ \left\{ \begin{array}{l}{{a}_{1}}=-3\\{{a}_{2}}=2\\{{a}_{n}}=3{{a}_{{n-1}}}-4{{a}_{{n-2}}};\,\,n>2\end{array} \right\}$.

We have the first two terms, and we need to get the third by using those first two terms, given the $ {{a}_{n}}$ rule. Note that $ {{a}_{n-1}}$ means one term back and $ {{a}_{n-2}}$ means two terms back from the present term. Therefore, the sequence is:

$ \begin{array}{l}\,\,\,-3,\,\,\,2,\,\,3\left( 2 \right)-4\left( {-3} \right),\,\,…\\=-3,\,\,2,\,\,18,\,\,3\left( {18} \right)-4\left( 2 \right),\,…\\=-3,\,\,2,\,\,18,\,\,46,\,\,3\left( {46} \right)-4\left( {18} \right),…\\=-3,\,\,2,\,\,18,\,\,46,\,\,66,…\end{array}$.

Arithmetic Sequences

Arithmetic sequences are those where the difference between the terms is always the same. They can be defined recursively as $ {{a}_{1}}=a;\,\,{{a}_{n}}={{a}_{{n-1}}}+\,\,d,\,\,\text{for}\,\,\,n>1$. Here, “$ a$” is the first term, and “$ d$” is the common difference. Arithmetic sequences are somewhat like linear equations (with the difference between terms like a slope), except sequences in general are discreet (just points) instead of continuous (a whole line).

The way I like to tell if a sequence is arithmetic is to see if “second term  – (minus) first term” equals “third term  –  second term” equals “fourth term  –  third term”, and so on. Then, once we have this number that is always the same, we have the common difference ($ d$). Note that “$ d$” can be negative, a fraction, or a decimal. (“$ n$” has to always be a positive integer).

Formula for an Arithmetic Sequence

We saw above that the if difference between the terms is $ d$ for an arithmetic sequence, the explicit formula then will have a “$ dn$” in it (such as “$ 4n$” if $ d=4$). We won’t prove it here, but it turns out the explicit formula for an arithmetic sequence is the formula below, for each term $ {{a}_{n}}$. This makes sense since we always start out with the first term, and then we are adding the common difference $ d$ “$ n-1$” more times.

This is called the formula for a general term  for an arithmetic sequence, and you’ll want to memorize this formula.

Note that all arithmetic sequences diverge; they never get closer and closer to any one number.

Here are some problems:


It turns out that if we are given the value of two specific terms of a sequence (and what terms they are – the “$ n$”), we can derive the equation of that sequence. Subtract the $ n$’s and subtract the terms ($ {{a}_{n}}$)’s and then divide these two numbers to get $ d$, the common difference. This makes sense since we are multiplying by $ d$ to get the successive terms. Then, to get the first term ($ {{a}_{1}}$), use the formula $ {{a}_{n}}={{a}_{1}}+\left( {n-1} \right)d$, using one of the $ {{a}_{1}}$ and $ n$ combinations, and the $ d$ we just got. Pretty cool!

Again, think of the common difference $ d$ as a slope of this “linear equation”; we are finding this slope using our familiar slope formula for two points of the form $ \left( {n,{{a}_{n}}} \right)$ as change of $ y$’s over change of $ x$’s: $ \displaystyle d=\,\frac{{{{a}_{n}}\,\text{(second)}-{{a}_{n}}\text{(first)}}}{{n\text{(second)}-n\text{(first)}}}$

Here are some problems:

Geometric Sequences

Whereas arithmetic sequences are those where the difference between the terms is the same, geometric sequences are those where the quotient of the terms is always the same. They can be defined recursively as $ {{a}_{1}}=\,\,a;\,\,\,\,{{a}_{n}}=r{{a}_{{n-1}}},\,\,\text{for}\,\,\,n>1$. Here, “$ a$” is the first term, and “$ r$” is the common ratio. Note that “$ r$” can be negative, a fraction, or a decimal. (“$ n$” has to always be a positive integer).

The way I like to tell if a sequence is geometric is to see if “second term /(divided by) first term” equals “third term/second term” equals “fourth term/third term”, and so on. Then, once we have this number that is always the same, we have the common ratio ($ r$). Geometric sequences are somewhat like Exponential Functions (with the common ratio like an exponential base), except sequences in general are discreet (just points) instead of continuous (a whole line).

You might be asked to find the geometric means in a geometric sequence; in this context, they are the middle terms in the sequence. For example, the geometric means between 1 and 8 (with a common ratio “$ r$” of 2) are 2 and 4, since $ 1\times 2=\underline{2}\times 2=\underline{4}\times 2=8$.

Formula for a Geometric Sequence

The explicit formula for a geometric sequence is the formula below, for each term. This makes sense since we always start out with the first term, and then we are multiplying the common ratio $ r$ “$ d$” more times.

This is called the formula for a general term  for a geometric sequence, and you’ll want to memorize this formula.

What this says is the $ n$^th term of an arithmetic sequence is the first term times $ r$ (common ratio) raised to the ($ d$).

IMPORTANT NOTE:
The above explicit formulas assume that the sequence begins at $ n=1$. Some textbooks show sequences starting with $ n=0$, and thus the Geometric Explicit formula is $ {{a}_{n}}=a{{\left( r \right)}^{n}}$, where $ a$ is the term where $ n=0$. This can be confusing, but the formula above will work if you assume the first term is in the first place (not place 0).

Here are some problems:


Just like for arithmetic sequences, it turns out that if we are given the value of two specific terms of a sequence (and what terms they are – the “$ n$”), we can derive the equation of that sequence. Subtract the $ n$’s (to get an $ x$) and divide the terms ($ {{a}_{n}}$ )’s (to get a $ y$) and then take the $ x$’th root of $ y$  to get $ r$, the common ratio. (We have to watch even roots, as shown in an example below). This makes sense since we are raising $ r$ to successive powers to get the successive terms. This way we can get the geometric means (the terms in between known terms) of a geometric sequence. Then, to get the first term ($ {{a}_{1}}$ ), use the formula $ {{a}_{n}}={{a}_{1}}{{\left( r \right)}^{{n-1}}}$, using one of the $ {{a}_{1}}$ and $ n$ combinations, and the $ r$ we just got. Pretty cool!

Here are some problems:


Here’s one more type of problem you may see:

Writing Formulas

Now let’s try to distinguish between arithmetic and geometric sequences, and also write formulas for those sequences that are neither arithmetic nor geometric. Remember again to look for second – first = third – second, and so on, for arithmetic sequences, and second / first = third / second, and so on, for arithmetic sequences. If neither of these work, look for squares or cubes in a row, added to or subtracted by a certain number.

Here’s another trick, if it appears the sequence is a quadratic (use this example:  3, 6, 10, 15, 21, …):


Of course, it might easier to just do a regression on your calculator with the sequence: use the $ n$ for $ x$-values and $ {{a}_{n}}$ for $ y$ values. We did regressions like this here in the Scatterplots, Correlation, and Regression section.

Here are some problems; we’ll be working again with explicit formulas (as opposed to recursive):

Arithmetic Series

When we add up some or all of the terms of a sequence, we have a series. When we add up the terms of an Arithmetic Sequence, we have an Arithmetic Series.

For example, the formula for the arithmetic sequence with $ {{a}_{1}}=1$ and $ d=2$ is $ {{a}_{n}}=1+\left( {n-1} \right)2=2n-1$. For $ n=1\,\text{to }5$, this sequence is 1, 3, 5, 7, 9. An arithmetic series is the sum of these elements, or 1 + 3 + 5 + 7 + 9. This partial sum (since we are only adding up 5 terms) is 1 + 3 + 5 + 7 + 9 = 25.

Here’s an example of how we might use an arithmetic series. Let’s say we have a theater that we are building and the first row has 20 seats, and every row has 2 more seats than the row before We want 100 rows of the theater, and we need to order the total number of seats in the theater. Do you see how we could get the sum of an arithmetic series of 100 terms (with common difference 2) to find out how many seats to order? Math is useful!

Summation Notation

An arithmetic series is often represented in a more compact way called summation notation or sigma notation. This is also called a summation formula.

Using the example above of the series 1 + 3 + 5 + 7 + 9, we have $ \sum\limits_{{k=1}}^{5}{{\,1+\left( {k-1} \right)2}}=\sum\limits_{{k=1}}^{5}{{\,2k-1}}$ (we usually use the variable “$ k$”). The sigma sign, or $ \sum{{}}$ symbol means take the equation after it, start with the lower bound, or what’s at the bottom of the sigma sign (1 in this case), plug in for $ n$, and add up everything until you get to the upper bound, or what’s at the top of the sigma sign (5 in this case; there will be a sum of 5 terms).

For the example with the theater seats above, where we start out with 20 seats, and add 2 seats to each row up to 100 rows, the summation notation would be $ \displaystyle \sum\limits_{{k=1}}^{{100}}{{\,20+\left( {k-1} \right)}}2=\sum\limits_{{k=1}}^{{100}}{{\,2k+18}}$. As another example, in the summation $ \displaystyle \sum\limits_{{k=0}}^{6}{{5k}}$, there are 7 terms that are added together ($ k=0,1,2,3,4,5,6$). The sum is:

$ \begin{array}{l}5\cdot 0+5\cdot 1+5\cdot 2+5\cdot 3+\\\,\,\,\,\,\,\,\,5\cdot 4+5\cdot 5+5\cdot 6=105\end{array}$.

Here are some problems:


Here are some problems where we use some very useful summation formulas (we see these in calculus in the Riemann Sums and Area by Limit Definition section here) to simplify our answers. These are typically given so you don’t have to memorize them:

$ \displaystyle \begin{align}\sum\limits_{{i=1}}^{n}{{c=cn\,\,(c\text{ is a constant)};\,\,\,\,\,\,}}\sum\limits_{{k=1}}^{n}{{k=\frac{{n\left( {n+1} \right)}}{2}\,\,\,\,}}\\\sum\limits_{{k=1}}^{n}{{{{k}^{2}}=\frac{{n\left( {n+1} \right)\left( {2n+1} \right)}}{6};\,\,\,\,}}\sum\limits_{{k=1}}^{n}{{{{k}^{3}}=\frac{{{{n}^{2}}{{{\left( {n+1} \right)}}^{2}}}}{4}\,\,}}\end{align}$

Here are some problems:

Formula for an Arithmetic Series

Below is the partial sum formula for an arithmetic series. Note that we can never get the total or infinite sum of an arithmetic series, since this sum would just go on forever (extremely large or extremely small). We’ll talk how this will be a Limit later.

Notice how the we’re using the formula for the $ n$^th term of an arithmetic sequence ($ {{a}_{n}}={{a}_{1}}+\left( {n-1} \right)d$) to get the second formula. (Personally, I prefer not to use the second formula, but just memorize the first, since we already have memorized the $ {{a}_{n}}$  formula.)

It’s pretty straightforward to use this formula to solve partial sum arithmetic series problems:

Geometric Series

Just like we saw that an arithmetic series is the sum of an Arithmetic sequence, Geometric Series are the sum of Geometric Sequences.

For example, for the geometric sequence $ {{a}_{n}}=2{{\left( 3 \right)}^{{n-1}}}$, for $ n=1\,\,\,\text{to }5$, we have 2, 6, 18, 54, 162. We would write this series as 2 + 6 + 18 + 54 + 162. Again, another way to write this is to use the summation formula $ \sum\limits_{{k=1}}^{5}{{\,2{{{\left( 3 \right)}}^{{k-1}}}}}$. (The generic summation formula is $ \sum\limits_{{k=1}}^{n}{{\,{{a}_{1}}{{{\left( r \right)}}^{{k-1}}}}}$).

Here’s an example of why we might want to use a geometric series. Let’s say we are getting a 5% raise for the next five years, and we want to know the total amount of money we will make for these five years. We make $50,000 now. Our common ratio is 1.05 (do you see how we multiply this by what we make every year to get our new salary?). Thus, we need the sum of a geometric series of 5 terms (with common ratio 1.05) to get the total amount we’ll make in these five year, or $ \sum\limits_{{k=1}}^{5}{{\,50000{{{\left( {1.05} \right)}}^{{k-1}}}}}$.

Finite Geometric Series

For geometric series, we have different formulas, depending on whether the series ends at a certain point (finite), or goes on forever (infinite). It turns out that we may have an infinite geometric series that actually ends up at a number; this is said to converge to this number, since adding each term makes the series get closer to this number. We’ll talk about this below.

Here is the partial sum formula for the finite geometric series (one that doesn’t go to infinity):

This looks really complicated, but it’s really not too bad; if we are solving for the $ n$^th generic term $ {{S}_{n}}$, we just plug in the first term ($ {{a}_{1}}$) and $ r$ (common ratio), and our answer has an “$ n$” in it. If we are solving for an actual number, our answer won’t have an “$ n$” in it. It’s actually a little easier than the arithmetic series formula, since we don’t need to find $ {{a}_{n}}$ first.

Note: We have to be careful to compute $ {{r}^{n}}$  first, before subtracting this from 1, and then multiply this difference by $ {{a}_{n}}$, and then divide the result by $ \left( {1-r} \right)$. We can’t simply cancel out the $ \left( {1-r} \right)$ and the $ \left( {1-{{r}^{n}}} \right)$, unless $ n=1$.

Here are some problems:

Infinite Geometric Series

A geometric series is infinite if it’s upper bound (the number on the top of the sigma sign) is infinity, or $ \infty $. Thus, an infinite series would look like this: $ \sum\limits_{{k=1}}^{\infty }{{2{{{\left( {.5} \right)}}^{{k-1}}}}}$.

It turns out that the equation for an infinite series is much easier than that of a finite series, but there’s one caveat: we can only get an answer (meaning, the series converges), if the common ratio, or $ r$, is between –1 and 1 (not including –1 and 1). This is the same thing as writing the restriction of $ r$ this way: $ \left| r \right|<1$. Note that the series diverges if  $ \left| r \right|\ge 1$; we can’t find an answer to the infinite series.

This makes sense since if $ r$ is a fraction, we’ll be adding a term that is getting smaller and smaller, so we should finally arrive at a number if we theoretically go to infinity (called the “limit”).

Here is the formula:

Here are some problems; determine whether the geometric series converges or diverges; if it converges, find the sum:

Summary of Formulas for Sequences and Series

Here is a summary of the main formulas for Sequences and Series:

Applications of Sequences and Series

Sequence and Series are very useful in many applications; in fact, with geometric sequences, especially when we’re dealing with growth or decay (like with money), we’ll see that they equations look a lot like some of the exponential equations we worked with here in the Exponential Functions section.

The main thing to remember about word problems with sequences and series is that when we want an amount for a single thing, such as a particular row, year, for example, we use a sequence. When we want to know a total amount, such as money or rows, we want to use a series (which is a sum).

Here are some problems:


Here are a few more problems where it might be helpful to draw pictures:

Sequences and Sums on the Graphing Calculator

We can use the TI Graphing Calculator to create sequences and determine the sum of sequences (series). Use SEQ and SUM in the Catalog list 2^nd 0), or, more easily, with 2^nd STAT (LIST) OPS 5 (or seq) and 2^nd STAT (LIST) MATH 5 (or sum), as in the following. Note that the calculator will either have you fill in the steps or, if you have an older operating system, you may have to enter the arguments on one line.

For seq, enter the expression, using the X,T,θ,n button, the variable (hit  X,T,θ,n again), starting and ending values of the variable, and (optional) 1 if you want all the values – which you probably will. With the new operating systems, using the MATHPRINT mode, you will be prompted for these values.

For sum, the calculator is looking for a list, so you can use it in conjunction with the seq command, as shown below. With the newer operating systems, you can get a summation symbol where you can enter the variable, the beginning and ending values, and the expression (you don’t have to use the seq command), like in the last screen. You can  also use MATH summation (MATH 0) or the shortcut ALPHA WINDOW 2 to do this.

Note that if you want the sum of an infinite series, you can put a large number in the upper bound, such as 999, and you should arrive at the solution, or pretty close!

You can even put a sequence in a LIST, like $ {{L}_{1}}$, by using the seq command and then using STO and then the name of the list:

Learn these rules, and practice, practice, practice!

For Practice: Use the Mathway widget below to try a Sequences and Series problem. Click on Submit (the blue arrow to the right of the problem) and click on Identify the Sequence to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to Binomial Expansion – you are ready!