In this explainer, we will learn how to represent a sequence as a function of a positive integer variable called an index .
Definition: Sequence
A sequence is an ordered list of terms.
Terms are usually named either or , where or is the index.
It is important to note that when discussing sequences, the use of the word “index” is different in meaning to that of “exponent.” In sequences, the index number refers to the position of the term in the sequence.
A sequence can have the index, , as or .
When a sequence begins with , the terms would be written as Here, is referred to as the “zeroth” term.
When a sequence begins with , the terms would be written as
If we consider the term labeled as ; the index is 3, and this would be the third term of the sequence. This is clear when the sequence has index , and is the third value in the sequence:
However, we note that, in a sequence with , with the terms , the term with index 3, would be the fourth value in the sequence.
Terms in a sequence can be given as a list or defined by a rule often linked to the index.
There are a number of different types of sequences. The first type of sequence is one in which the difference between any pair of successive terms is constant. For example, if we look at the sequence we notice that there is a difference of 3 between each pair of consecutive terms.
We can examine the pattern to find a rule for any term in the sequence by writing each term in terms of the first term.
| Index | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value of Term | 5 | 8 | 11 | 14 |
| Value in terms of First Term | 5 |
For example, the third term, 11, is the first term, 5, plus 6. We can then consider that we can also write each term as an expression related to the difference between terms and an increasing index.
| Index | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value of Term | 5 | 8 | 11 | 14 |
| Value in terms of First Term | 5 | |||
| Value in terms of Difference and Index |
Each value in the parentheses is one less than the index value. We could write any th term in the sequence, in words, as
Notice that this rule for the th term even works for the first term, as the simplifies to 0, and using the th term formula would leave us with just the value for the first term.
This rule for the th term of a sequence works for any arithmetic sequence, and we define this more formally below.
Definition: Arithmetic Sequence
An arithmetic sequence is a sequence that has a fixed, or common, difference between any two successive terms.
An arithmetic sequence of index has a general, or th, term of where is the term with position number 1 and is the common difference.
In the first question, we will need to establish which type of sequence we have and then find its th term.
Example 1: Finding the Common Difference and General Term of an Arithmetic Sequence
In any sequence pattern, if the difference between any two successive terms is a fixed number, , then this is an arithmetic sequence.
Consider the sequence , and then answer the following questions.
- Is the sequence arithmetic?
- What is the value of ?
- What is the general term of this sequence with ?
Answer
Part 1
We are reminded that an arithmetic sequence has a common difference between successive terms; so, to consider if this term is arithmetic, we check if there is a common difference.
The difference between the first and second terms, 1 and 4, can be calculated by subtracting the first term from the second. Therefore, we have
The difference between the second and third terms is calculated by
Finally, the difference between the third and fourth terms is
We can visualize these differences below.
As the difference between any pairs of successive terms is a fixed number, we give the answer: yes, the sequence is arithmetic.
Part 2
The value of is the common difference, and it is the value we worked out in the first part of this question. Therefore, is 3.
Part 3
Let us now see how we can find the general term, or th term, of the sequence. As we are given that the index, , must be an integer with , this means the first index of the sequence will be .
We can consider the terms and their indexes listed below.
| Index | 0 | 1 | 2 | 3 | |
|---|---|---|---|---|---|
| Term | 1 | 4 | 7 | 10 |
Looking at the table, we can see that the sequence begins with a “zeroth” term; that is, . The term with index 1, , is in fact the next term with a value of 4.
To find the th term of a sequence, we can apply the rule that, for an index , where is the term with index 1 and is the common difference.
For the sequence , we have established that the difference and the term with index 1 is 4, so . We can substitute these into the formula above to give
Multiplying each term in the parentheses, , by 3, we have
Finally, we can simplify by collecting like terms to give
Thus, we have found that the general term of the sequence , with , is
We can check our answer by substituting the index values, , into the general term, .
When , we have
When , we have And when , we calculate
As the values 1, 4, and 7 were the given values of the first terms of the sequence, we have confirmed that the general term must be
Example 2: Finding the General Term of an Arithmetic Sequence
Which of the following expressions can be used to find the th term of the given sequence, where represents the position of a term in the sequence?
| Position | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value of Term | 5 | 8 | 11 | 14 | 17 |
Answer
We can begin answering this question by finding the difference between terms. If there is a common difference between terms, then the sequence is arithmetic.
We observe that, between any two successive terms, the difference is 3. In order to find the th term, we recall that an arithmetic sequence of index has a general, or th, term of where is the term with position number 1 and is the common difference.
In this sequence, the difference, , is 3, and the term with position number 1, , is 5. Thus, substituting these values into gives
Therefore, we can give the th term, , as
In the first two examples, we looked at how to interpret and represent arithmetic sequences. The next type of sequence we are interested in is a geometric sequence. A geometric sequence has a constant ratio between successive terms. For example, consider the sequence . Each term can be multiplied by a value of 4 to get the term immediately following it.
We could think of this sequence in the following way.
| Index | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value of Term | 1 | 4 | 16 | 64 |
| Value in terms of First Term and Ratio | 1 |
We can further simplify this final row of values by writing these in terms of exponents of the ratio, 4.
| Index | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value of Term | 1 | 4 | 16 | 64 |
| Value in terms of First Term and Ratio |
We can now relate this exponent to the index number. As each exponent is one less than the index, any th term in the sequence can be written in words as
Defining as the first term and as the common ratio, we could write the terms in a geometric sequence as
Extending this sequence, we find that the th term can be written as
This definition is formalized below.
Definition: Geometric Sequence
A geometric sequence is a sequence that has a fixed, or common, ratio between any two successive terms.
A geometric sequence of index has a general, or th, term of where is the first term and is the common ratio.
We can find the common ratio of a geometric sequence by dividing any term in a sequence by the term immediately preceding it; that is, where , or where .
We will now look at an example where we will need to establish whether a sequence is geometric or not and then find its th term.
Example 3: Finding the Common Ratio and General Term of a Geometric Sequence
In a geometric sequence, the ratio between any two successive terms is a fixed ratio .
Consider the sequence
- Is this sequence geometric?
- What is the value of ?
- What is the general term of this sequence?
Answer
Part 1
In a geometric sequence, as the ratio between terms is a constant, this means that we can check if a sequence is geometric by finding the ratio between terms.
To find the ratio between any two terms, we divide any term by the term preceding it. This can also be written as for any index, , in the sequence, where . We can list the index and term value for each term in the sequence.
| Term Index | |||||
|---|---|---|---|---|---|
| Term |
To find the ratio between the terms with position numbers 1 and 2, and , we calculate
When we divide fractions, this is equivalent to multiplying by the reciprocal of the second fraction. So, we evaluate
The ratio between the first and second terms is . We now need to check if this is the same as the ratio between the terms with position numbers 2 and 3. We can calculate divided by :
We can check the final two terms to find the ratio between them. In the same way, we see
As all the given terms have a fixed ratio between successive terms, we can answer the question: yes, the sequence is geometric.
Part 2
The value of the common, or fixed ratio, is defined as . In the first part of the question, we saw that the ratio between terms is . Therefore, has the value .
Part 3
To find the general term of a sequence, we can recall that, for an index , this is given by where is the term with position number 1 and is the common ratio.
For this sequence, and . Substituting these values into the formula above, we have
The values that we used here for the index were integer values beginning with , and we must note this as part of our answer, as . Therefore, we can give the general term of the sequence as
Notice that we could have further simplified this solution using one of the rules of exponents, , to give the equivalent general term as
One way to check our solution is by substituting in the values , and 4 into either form of the general term. As both forms produce the same terms of the sequence as , we have verified our answer for the general term.
In the following question, we will see how we can generate specific terms in a sequence given its th term.
Example 4: Finding the First Three Terms of a Sequence When Given the 𝑛th Term
Find the first 3 terms of the sequence whose general term .
Answer
The general term of a sequence allows us to find any term of that sequence according to the value of its index, . For example, if we wanted to find the tenth term, for values of the index , we would find the term by substituting into the general term.
To find the first term, we can substitute into the general term; for the second term, we can substitute ; and for the third term, .
So, for the first term, let in the general term . This gives
Therefore, the term with position number 1 is .
To find the second term, we substitute into the general term . So, we have
For the third term, we substitute into the general term , giving us
Now, we know that the first term is , the second is , and the third is . Therefore, we can give the answer that the first 3 terms of the sequence are
In the final question, we will find the th term of a different kind of sequence. To do so, we can take a logical approach, breaking down the fractions in the terms of the sequence to allow us to create a general term for the sequence.
Example 5: Finding the 𝑛th Term of an Unknown Sequence
Consider the sequence .
Which of the following is the general term of this sequence such that ?
Answer
To find the general term, it may be useful to begin by establishing if this is a geometric, arithmetic, or another type of sequence. We can recall that an arithmetic sequence has a common, or fixed, difference between terms. However, when we inspect the sequence, by subtracting any term from the term immediately after it, we notice that the difference is not fixed between all terms.
This means that the sequence is not arithmetic.
A geometric sequence has a common ratio between all terms. When we consider the sequence and divide any term by the term before it, the ratio between terms is not fixed.
Therefore, the sequence is not a geometric one either.
Another way to approach this question is to use some logic and look again at the terms. We can consider the terms alongside their index values. As we are given that the index, , must have the values , we consider the index to begin with .
| Term Index | |||||
|---|---|---|---|---|---|
| Term | 1 | 1 |
Let us look a little closer at the terms with indexes 0 and 1, and , respectively. Let us consider if these terms are fractions that simplified to 1. Any fraction that simplifies to 1 must have the numerator and denominator the same; that is, . We may even posit that is the value . Let us see how we can use this, and consider the numerators and denominators separately, in order to find a fractional value for .
| Term Index | |||||
|---|---|---|---|---|---|
| Term | 1 | 1 | |||
| Numerator | 1 | 3 | 4 |
We may notice that if the numerator of a fractional value for was 2, then there would be a pattern of 1, 2, 3, 4 on the numerator. As this value as a fraction, , would have to simplify to 1, the denominator would also have to be 2.
| Term Index | |||||
|---|---|---|---|---|---|
| Term | 1 | ||||
| Numerator | 1 | 2 | 3 | 4 | |
| Denominator | 1 | 2 | 4 | 8 |
Now, the numerators form an arithmetic sequence with a common difference of 1, and the denominators form a geometric sequence with a common ratio of 2. We can now write these in terms of the index, .
For the numerators, each value in the term is one more than the index value; for example, when , the numerator of the term is 1. We can apply the rule to find the general sequence of an arithmetic sequence, where is the term with index 1 and is the common difference.
Substituting and into gives
Thus, we can write the general term for the numerator as
Similarly, we can use the general term of a geometric sequence to find the th term for the denominators.
The general term, , of a geometric sequence is given by where is the term with index 1 and is the common ratio.
Substituting and into the general term, we have
Therefore, the general term for the denominator will be .
Finally, for the general term, we can combine the rules for the numerators and denominators. The general term is option C:
We will now recap the key points developed in this explainer.
Key Points
- A sequence is made up of an ordered list of terms.
- Terms may be given as , where is the index.
- The index, , will be integer values beginning with either 0, when , or 1, when .
- Arithmetic sequences have a fixed difference between any two successive terms. The general term, or th term, of an arithmetic sequence is given by where is the term with position number 1 and is the common difference.
- Geometric sequences have a fixed ratio between any two successive terms. The general, or th term, of a geometric sequence is given by where is the term with position number 1 and is the common ratio.
- Sequences can be arithmetic, geometric, or neither. We can often consider fractional sequences as being composed of separate general terms for the numerator and denominator.
