# What are sequences in math?

An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence then subtract it by the previous one, and the result is always the same or constant then it is an arithmetic sequence.

The constant difference in all pairs of consecutive or successive numbers in a sequence is called the common difference, denoted by the letter d. We use the common difference to go from one term to another. How? Take the current term and add the common difference to get to the next term, and so on. That is how the terms in the sequence are generated.

• If the common difference between consecutive terms is positive, we say that the sequence is increasing.
• On the other hand, when the difference is negative we say that the sequence is decreasing.

## Illustrative Examples of Increasing and Decreasing Arithmetic Sequences

Here are two examples of arithmetic sequences. Observe their common differences.

With this basic idea in mind, you can now solve basic arithmetic sequence problems.

### Examples of How to Apply the Concept of Arithmetic Sequence

Example 1: Find the next term in the sequence below.

First, find the common difference of each pair of consecutive numbers.

Since the common difference is 8 or written as d=8, we can find the next term after 31 by adding 8 to it. Therefore, we have 31 + 8 = 39.

Example 2: Find the next term in the sequence below.

Observe that the sequence is decreasing. We expect to have a common difference that is negative in value.

To get to the next term, we will add this common difference of d=-7 to the last term in the sequence. Therefore, 10 + left( { – 7}
ight) = 3.

Example 3: Find the next three terms in the sequence below.

Be careful here. Don’t assume that if the terms in the sequence are all negative numbers, it is a decreasing sequence. Remember, it is decreasing whenever the common difference is negative. So let’s find the common difference by taking each term and subtracting it by the term that comes before it.

The common difference here is positive four left( { + ,4}
ight) which makes this an increasing arithmetic sequence. We can obtain the next three terms by adding the last term by this common difference. Whatever is the result, add again by 4, and do it one more time.

Here’s the calculation:

The next three terms in the sequence are shown in red.

Example 4: Find the seventh term (7th) in the sequence below.

Sometimes you may encounter a problem in an arithmetic sequence that involves fractions. So be ready to use your previous knowledge on how to add or subtract fractions.

Also, always make sure that you understand what the question is asking so that you can have the correct strategy to approach the problem.

In this example, we are asked to find the seventh term, not simply the next term. It is a good practice to write all the terms in the sequence and label them, if possible.

Now we have a clear understanding of how to work this out. Find the common difference, and use this to find the seventh term.

Finding the common difference,

Then we find the 7th term by adding the common difference starting with the 4th term, and so on. Here’s the complete calculation.

Therefore, the seventh term of the sequence is zero (0). We can write the final answer as,

Example 5: Find the color{red}{35^{th}} term in the arithmetic sequence 3, 9, 15, 21, …

You can solve this problem by listing the successive terms using the common difference. This method is tedious because you will have to keep adding the common difference (which is 6) thirty-five times starting with the last term in the sequence.

You don’t have to do this because it is cumbersome. And not only that, it is easy to commit a careless error during the repetitive addition process.

If you decide to find the color{red}{35^{th}} term of the sequence using this “successive addition” method, your solution will look similar below. The “dot dot dot” means that there are calculations there but not shown as it can easily occupy the entire page.

• You might also be interested in:
• Arithmetic Sequence Formula
• More Practice Problems with the Arithmetic Sequence Formula
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## Number Sequences (solutions, examples, videos)

Related Topics: More Lessons for Arithmetic Math Worksheets How to Find The Next Term In A Number Sequence?

A number sequence is a list of numbers arranged in a row. Let us look at two examples below.

(i) 4, 6, 1, 10, 14, 5, …

(ii) 4, 7, 10, 13, ….

Number sequence (i) is a list of numbers without order or pattern. You cannot tell what number comes after 5.

Number sequence (ii) has a pattern. Do you observe that each number is obtained by adding 3 to the preceding number (i.e. the number just before it)?

In this section, we will only study number sequences with patterns .

Some other examples of number sequences are:

 Number Sequence Pattern 3, 6, 9, 12, … add 3 12, 17, 22, 27, … add 5 70, 65, 60, 55, … subtract 5 15, 19, 23, 27, … add 4 81, 27, 9, 3, … divide by 3

How to Complete Missing Terms In A Number Sequence?

• Each of the number in the sequence is called a term.
• In order to find the missing terms in a number sequence, we must first find the pattern of the number sequence.
• Example :
• Find the missing term in the following sequence:
• 8, ______, 16, ______, 24, 28, 32
• Solution:

To find the pattern, look closely at 24, 28 and 32. Each term in the number sequence is formed by adding 4 to the preceding number. So, the missing terms are 8 + 4 =12 and 16 + 4 = 20. Check that the pattern is correct for the whole sequence from 8 to 32.

1. Example :
2. What is the value of n in the following number sequence?
3. 16, 21, n, 31, 36
4. Solution:
5. We find that the number pattern of the sequence is “add 5” to the preceding number. So, n = 21 + 5 = 26

How to find the next term in a number sequence? The following video shows some examples of how to determine the next term in a number sequence. Examples: Find the next number 1. 1, 8, 15, 22, … 2. 1, 8, 64, 512, … 3. 1, 8, 27, 64, ….

4. 1, 8, 16, 15, …

• Show Step-by-step Solutions

The following diagrams give the formulas for Arithmetic Sequence and Geometric Sequence. Scroll down the page for examples and solutions. How to find the nth Term of an Arithmetic Sequence Example:

7, 9, 11, 13, 15, …

• Show Step-by-step Solutions

How to find the nth Term of a Geometric Sequence? Example:

5, 10, 20, 40, …

• Show Step-by-step Solutions

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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## Describing Sequences | Number Patterns

A sequence is an ordered list of items, usually numbers. Each item which makes up a sequence is called a “term”.

Sequences can have interesting patterns. Here we examine some types of patterns and how they are formed.

Examples:

• (1; 4; 7; 10; 13; 16; 19; 22; 25; ldots)
• There is difference of ( ext{3}) between successive terms.
• The pattern is continued by adding ( ext{3}) to the previous term.
1. (13; 8; 3; -2; -7; -12; -17; -22; ldots)

There is a difference of (- ext{5}) between successive terms.

The pattern is continued by adding (- ext{5}) to (i.e. subtracting ( ext{5}) from) the previous term.

1. (2; 4; 8; 16; 32; 64; 128; 256; ldots)
2. This sequence has a factor of ( ext{2}) between successive terms.
3. The pattern is continued by multiplying the previous term by 2.
• (3; -9; 27; -81; 243; -729; 2187; ldots)
• This sequence has a factor of (- ext{3}) between successive terms.
• The pattern is continued by multiplying the previous term by (- ext{3}).
1. (9; 3; 1; frac{1}{3}; frac{1}{9}; frac{1}{27}; ldots)
2. This sequence has a factor of (frac{1}{3}) between successive terms.
3. The pattern is continued by multiplying the previous term by (frac{1}{3}) which is equivalent to dividing the previous term by 3.

Some learners may see example 3 as (2^{1}; 2^{2}; 2^{3}; ldots) and see a pattern with the powers. You may choose to discuss this in class as a precursor to geometric series which will be introduced in Grade 12.

## Sequences

A sequence is a set of ordered numbers. For example, the sequence 2, 4, 6, 8, … has 2 as its first term, 4 as its second, etc. The nth term in a sequence is usually called sn. The terms of a sequence may be arbitrary, or they may be defined by a formula, such as sn = 2n.

In general, n starts at 1 for sequences, but there are times when it is convenient for n to start at 0, in which case the first term is s0. If we add up the first n terms of a sequence we get a partial sum, usually referred to as Sn (i.e., with a capital letter).

This device cannot display Java animations. The above is a substitute static image

 See About the calculus applets for operating instructions.

### 1. Arithmetic sequence

The applet shows the sequence defined by sn = 2 + 3(n – 1). This is called an arithmetic sequence and each term of the sequence is found by adding a constant amount (e.g., 3 in this example) to the preceeding element.

The general formula for an arithmetic sequence is sn = s1 + d(n – 1), where s1 is the first term and d is the common difference (i.e., the amount added to get the next term).

The partial sum of the first 10 terms is shown in the upper left corner of the graph, and you can change the number of terms by moving the max n slider or typing in the max n input box.

One of the issues that we are concerned with when working with sequences is what happens to the values of the terms when n heads to infinity.

In other words, does
have a value, or does sn head off to infinity or jump around as n gets big? If there is a limit, we say that the sequence converges or is convergent.

If this limit does not exist, the sequence diverges or is divergent. Obviously the arithmetic sequence diverges, because the terms keep getting bigger.

Select the second example from the drop down menu, showing a geometric sequence defined by

sn = 2n In a geometric sequence each term is a constant multiple of the previous term (the multiple here is 2). The general form of a geometric sequence is

sn = s1rn – 1 where r is the common ratio (i.e., the amount that each term is multiplied by to get the next term). Obviously, r = 1 and r = 0 are not useful cases (both just give a constant value for all terms). It is clear from the graph that the example sequence is divergent, because the terms keep getting bigger.

### 3. Another geometric sequence

Select the third example, showing another geometric sequence with a common ratio of 1/2. Does this one converge? The terms get closer and closer to zero, so this sequence does converge. Geometric sequences converge if the common ratio is between 0 and 1, and diverge if the common ratio is greater than 1.

### 4. Alternating geometric sequence

Select the fourth example, showing another geometric sequence with a negative common ratio. Note that the terms alternate on the positive and negative side of the axis. This sequence also converges towards 0, so we can extend our knowledge of geometric sequence convergence to say that the sequence converges if |r| < 1.

### Explore

You can experiment with your own sequences by typing in a rule, using n as the variable.

## Sequences – Sequences – AQA – GCSE Maths Revision – AQA – BBC Bitesize

• Number sequences are sets of numbers that follow a pattern or a rule.
• If the rule is to add or subtract a number each time, it is called an arithmetic sequence.
• If the rule is to multiply or divide by a number each time, it is called a geometric sequence.
• Each number in a sequence is called a term.
• A sequence which increases or decreases by the same amount each time is called a linear sequence.

The term to term rule of a sequence describes how to get from one term to the next.

### Example 1

Write down the term to term rule and then work out the next two terms in the following sequence.

3, 7, 11, 15, …

Firstly, work out the difference in the terms.

This sequence is going up by four each time, so add 4 on to the last term to find the next term in the sequence.

3, 7, 11, 15, 19, 23, …

To work out the term to term rule, give the starting number of the sequence and then describe the pattern of the numbers.

The first number is 3. The term to term rule is 'add 4'.

Once the first term and term to term rule are known, all the terms in the sequence can be found.

### Example 2

Write down the term to term rule and then work out the next two terms in the following sequence.

-1, -0.5, 0, 0.5, …

The first term is -1. The term to term rule is 'add 0.5'.

Question

What is the term to term rule and the next two terms of the sequence: 17, 14, 11, 8, …?

To work out the term to term rule, give the first term and then the pattern. The first term is 17, and the pattern is to subtract 3 each time, so the term to term rule is 'start at 17 and subtract 3'.

The next two terms of the sequence are 5 and 2, giving the sequence as:

Question

What are the next three terms of a sequence that has a first term of 1, where the term to term rule is multiply by 2?

The first term is given as 1. Each number that follows is double the number before.

## Sequences | Brilliant Math & Science Wiki

A sequence is an ordered set with members called terms.

Usually, the terms are numbers. A sequence can have infinite terms.

An example of a sequence is

1,2,3,4,5,6,7,8,… .1,2,3,4,5,6,7,8,dots.1,2,3,4,5,6,7,8,….

There are different types of sequences. For example, an arithmetic sequence is when the difference between any two consecutive terms in the sequence is the same. So,

5,14,23,32,41,505, 14, 23, 32, 41,505,14,23,32,41,50

is an arithmetic sequence with common difference 999, first term 555, and number of terms 6.6.6.

Another type of sequence is a geometric sequence. This is when the ratio of any two consecutive terms in the sequence is the same. For example,

2,6,18,54,1622, 6, 18, 54, 1622,6,18,54,162

is a geometric sequence with common ratio 333, first term 222, and number of terms 5.5.5.

In a sequence, it is conventional to use the following variables:

• aaa is the first term in the sequence.
• nnn is the number of terms in the sequence.
• Tn{ T }_{ n }Tn​ is the nthn^ ext{th}nth term in the sequence.
• Sn{ S }_{ n }Sn​ is the sum of the first nnn terms of the sequence.
• ddd is the common difference between any two consecutive terms (arithmetic sequences only).
• rrr is the common ratio between any two consecutive terms (geometric sequence only).

For example, if a series starts with 111 and has a common difference of 1,1,1, we have Sn=n(n+1)2.{ S }_{ n }= dfrac{n(n + 1)}{2}.Sn​=2n(n+1)​.

Similarly, for the series of squares 12,22,32,…,n2,1^2,2^2,3^2,dots,n^2,12,22,32,…,n2, we have Sn=n(n+1)(2n+1)6.{ S }_{ n } = dfrac{n(n + 1)(2n + 1)}{6}.Sn​=6n(n+1)(2n+1)​.

For the series of cubes 13,23,33,…,n3,1^3,2^3,3^3,dots,n^3,13,23,33,…,n3, we have Sn=(n(n+1)2)2.{ S }_{ n } = left(dfrac{n(n+1)}{2}
ight)^2.Sn​=(2n(n+1)​)2.

Some special types of sequences can be found in Arithmetic Progressions, Geometric Progressions, and Harmonic Progression.