In this explainer, we will learn how to write explicit and recursive formulas for arithmetic sequences to find the value of the term in an arithmetic sequence and how to find a termโs order given its value.

A sequence, , is an enumerated collection of numbers (or other objects) that usually follow a pattern. The individual elements in a sequence, for , are called the terms and are labeled by the index that tells us the position of the given term in the sequence.

Now, letโs recall the definition of an arithmetic sequence.

### Definition: Arithmetic Sequence

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers, , that has a constant nonzero common difference between any two consecutive terms: where is the term in the arithmetic sequence.

The arithmetic sequence in general can also be depicted as

In order to calculate the common difference of a given arithmetic sequence, we can subtract any term from the sequence by the term that immediately precedes it (e.g., we could subtract the second term from the third term or the first term from the second term in the sequence; either way will yield the same number for an arithmetic sequence). For example, if we have the sequence , we can clearly see a common difference between consecutive terms:

This sequence can be depicted as

As we may note from the definition, the recursive formula for the arithmetic sequence can be written as

In other words, a term in an arithmetic sequence is found by adding the common difference to the previous term. In order to define an arithmetic sequence, we need to be given or determine the common difference and the first term .

Now, letโs consider an example where we determine a term of an arithmetic sequence using the recursive relation.

### Example 1: Finding a Particular Term from the Recursive Formula of an Arithmetic Sequence

Fill in the blank: If the first term of an arithmetic sequence equals 16 and , then the fifth term equals .

### Answer

In this example, we want to determine the value of the fifth term of an arithmetic sequence, given a recursive formula and the first term.

We can work out the value of each term in the arithmetic sequence by using and substituting the values into the recursive formula:

Thus, the value of the fifth term equals 24.

If we denote the first term as for simplicity, the general form of an arithmetic sequence is

The second term of the arithmetic sequence is calculated by adding to the first term, , to obtain . The third term is the second term plus , to give :

In other words, we produce each term by adding the same number, , to the term that precedes it. Using the general form, we can write an explicit formula for the term as

In general, applying the recursive relation repeatedly, we can show that which allows us to determine the value of the term in the sequence, , from the term, :

Now, letโs consider a few examples of how to find the explicit formulas of given arithmetic sequences. In the next example, we will determine this from the first few terms.

### Example 2: Finding the General Term of an Arithmetic Sequence

Find, in terms of , the general term of the arithmetic sequence .

### Answer

In this example, we want to determine the general term of a given arithmetic sequence.

Recall that an arithmetic sequence is defined by a constant common difference, , between any two consecutive terms. The explicit formula for the term can be written in terms of the common difference and the first term, , as

We are given the first few values of the sequence, . Letโs first calculate the difference between consecutive terms:

Thus, we have a common difference , which confirms that we have an arithmetic sequence.

The general term for the given arithmetic sequence, using the common difference and first term , is

Hence, the general term of the sequence is .

Now, letโs consider an example where we determine the general term from a table with values starting from the sixth term and then evaluate the eighteenth term in the sequence.

### Example 3: Writing Algebraic Expressions from a Given Table Then Evaluating Them

Using the table, determine the expression that represents the value of each term as a function of its position. Then, find the value of the eighteenth term in the sequence.

Position | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|

Value of Term | 19 | 22 | 25 | 28 |

### Answer

In this example, we want to determine the general term of a sequence by using a given table showing the position and values of the terms in the sequence. In order to try and find some pattern in this sequence, let us consider the difference between successive terms:

We can see that each successive term can be obtained from the previous one by adding a common difference (). Recall that an arithmetic sequence is defined by a constant common difference, , between any two consecutive terms. Therefore, this must be an arithmetic sequence with common difference 3.

Recall that the explicit formula for the term of an arithmetic sequence can be written in terms of the common difference and the first term, , as

Now, we can determine the first term by substituting and into this formula:

Since we know that , we can substitute this into the formula to obtain

Therefore, the first term of the sequence is . The general term for the given arithmetic sequence, using the common difference and first term , is

Finally, we can determine the eighteenth term in the sequence by substituting to find

In the next example, we will determine the explicit formula of an arithmetic sequence where the terms are expressed in terms of two parameters. Using the explicit formula, we will then determine the nineteenth term in the sequence.

### Example 4: Finding the Value of a Term in a Given Arithmetic Sequence

Find in the arithmetic sequence .

### Answer

In this example, we want to find the value of a term in a given arithmetic sequence.

Recall that an arithmetic sequence is defined by a constant common difference, , between any two consecutive terms. The explicit formula for the term can be written in terms of the common difference and the first term, , as

Letโs first calculate the difference between consecutive terms:

The general term for the given arithmetic sequence, using the common difference and the first term , is

Finally, we can determine , the nineteenth term in the sequence, by substituting to find

Now, letโs consider an example where we determine the general term of an arithmetic sequence that satisfies particular conditions for particular terms in the sequence.

### Example 5: Finding the General Term of an Arithmetic Sequence under a Certain Condition

Find the general term of the arithmetic sequence that satisfies the relations and .

### Answer

In this example, we want to find the general term of an arithmetic sequence that satisfies particular conditions.

Recall that an arithmetic sequence is defined by a constant common difference, , between any two consecutive terms. The explicit formula for the term can be written in terms of the common difference and the first term, , as

Using this explicit formula with the condition , we have

Similarly, for the condition , we have

If we substitute the first equation , we obtain

Thus, we have two simultaneous equations:

Subtracting the second equation from the first, we obtain

We can now obtain the first term from either of the equations:

Therefore, the general term of the arithmetic sequence that satisfies the relations is

As we have seen so far, to determine a specific value of the term in an arithmetic sequence, we have to substitute the given value of in the explicit formula (i.e., for the 5th term, we substitute ).

But what if we want to do the reverse? In other words, for a given value in a sequence, we want to determine the value of the index , also known as the order of the term, which is the position where the value occurs in the sequence such that the explicit formula for gives this value. For a given value , the termโs order in an arithmetic sequence may be found by making the subject from the explicit formula:

For example, if we have an arithmetic sequence defined by , we can determine the order of the term equal to 92 (i.e., such that ) by solving the following equation for :

Thus, the order of the term 92 in the sequence is and thus .

In the next example, we will determine the order of a particular value in an arithmetic sequence after finding the general term.

### Example 6: Finding the Order of a Term in a Sequence given Its Value

Find the order of the term whose value is 112 in the sequence .

### Answer

In this example, we want to determine the order of a given term in an arithmetic sequence.

Recall that an arithmetic sequence is defined by a constant common difference, , between any two consecutive terms. The order of a value that occurs in an arithmetic sequence is the position or value of where we obtain the particular value from the explicit formula for .

The explicit formula for the term can be written in terms of the common difference and the first term, , as

Letโs first calculate the difference between consecutive terms:

This confirms that we indeed have an arithmetic sequence. The general term for the given arithmetic sequence, using the common difference and the first term , is

To determine the order of the term 112 in the arithmetic sequence, we want to determine the value of with or such that

Therefore, the order of the term 112 is and thus .

Now, letโs consider an example where we have to evaluate the first three terms of a given arithmetic sequence defined in terms of an explicit formula and then determine the order of a particular value and the first term whose value is more than a given number.

### Example 7: Finding the Arithmetic Sequence and the Order of Terms under a Certain Condition given the General Term of This Sequence

Find the first three terms in the sequence that has general term and . Then, find the order of the term whose value is 157 and the order of the first term whose value is more than 100.

### Answer

In this example, we want to determine the first three terms of an arithmetic sequence defined in terms of an explicit formula, then find the order of another term and the order of the first term whose value is greater than a particular value.

Recall that an arithmetic sequence is defined by a constant common difference between any two consecutive terms. The first three terms can be found from the general term by substituting , 2, and 3:

Thus, the first three terms are 31, 45, 59.

The order of a value that occurs in an arithmetic sequence is the position or value of where we obtain the particular value from the explicit formula for .

For the order of the term 157 in the arithmetic sequence, we want to determine the value of with or such that

Therefore, the order of the term 157 is and thus .

The order of the first term whose value is more than 100 can be found by solving the inequality for the smallest integer value of :

Hence, the smallest integer value of is , which is the order of the smallest term whose value is more than 100. In particular, we have .

In the next example, we will find the general term of an arithmetic sequence that satisfies particular conditions and then use it to determine the order and value of the first negative term in the sequence.

### Example 8: Finding the General Term of an Arithmetic Sequence and the Order and Value of the First Negative Term under a Given Condition

Find the term of the arithmetic sequence given that and . Then find the position and value of the first negative term in the sequence.

### Answer

In this example, we want to find the general term of an arithmetic sequence that satisfies particular conditions and then the order of the value of the first negative term in the sequence.

Using this explicit formula with the condition , we have

Similarly with , using , we obtain

The general term for the given arithmetic sequence, using the common difference and the first term , is

The position or order of the first negative term in the sequence can be found by solving the inequality for the smallest integer value of :

Thus, the smallest integer value is , which is the order of the first negative term in the sequence with value .

The number of terms in an arithmetic sequence, , is equivalent to the order of the last term, , in the sequence, which is .

In the final example, we will determine the value of an unknown parameter appearing in the terms of a given arithmetic sequence and then determine the total number of terms in the sequence.

### Example 9: Finding the Number of Terms in a Given Arithmetic Sequence

Find and the number of terms in the arithmetic sequence .

### Answer

In this example, we want to determine an unknown parameter appearing in the terms of an arithmetic sequence and then find the total number of terms in the sequence.

Recall that an arithmetic sequence is defined by a constant common difference, , between any two consecutive terms. We can use this property to determine the value of the unknown by calculating the difference between the second and first term, and then the last and second to last term,

Since the difference between two consecutive terms in an arithmetic sequence is constant, we can equate these two expressions for the common difference to obtain a single equation in terms of :

Thus, we have , which we can substitute into the terms of the given arithmetic sequence to obtain the sequence . This sequence has a common difference of .

The explicit formula for the term can be written in terms of the common difference and the first term, , as

The number of terms is equivalent to the order of the last term in the sequence, which we can determine from the explicit formula. For the given sequence, the general term is

Hence, the number of terms or the order of the last term, 40, in the arithmetic sequence can be found by solving , which gives .

Letโs recap the key points in this explainer.

### Key Points

- An arithmetic sequence is a sequence of numbers, , that has a constant nonzero common difference between any two consecutive terms:
- This formula can also be used to find the next terms in an arithmetic sequence from the common difference using the recursive relation: That is, each term in an arithmetic sequence is found by adding the common difference to the previous term.
- In general, applying the recursive relation repeatedly, we can show that which allows us to determine the value of the position in the sequence, , from the value.
- If we denote the starting value as for simplicity, the general form of an arithmetic sequence is From this general form, we also have an explicit formula for any term in the sequence: We can determine the explicit formula for an arithmetic sequence by identifying the first term and the common difference.
- A given arithmetic sequence may be defined in terms of a set of numbers, , the recursive formula with a given first term, or an explicit formula.
- For a given value in a sequence, , to determine the value of , the position of the term in the sequence also known as the order of the term, we have the formula