3.10 Arithmetic Sequences

Learning Objectives

After completing this section, you should be able to:

Identify arithmetic sequences.
Find a given term in an arithmetic sequence.
Find the $n$ th term of an arithmetic sequence.
Find the sum of a finite arithmetic sequence.
Use arithmetic sequences to solve real- world applications

As we saw in the previous section, we are adding about 2.5 quintillion bytes of data per day to the Internet. If there are 550 quintillion bytes of data today, then there will be 552.5 quintillion bytes tomorrow, and 555 quintillion bytes in 2 days. This is an example of an arithmetic sequence. There are many situations where this concept of fixed increases comes into play, such as raises or table arrangements.

Identifying Arithmetic Sequences

A sequence of numbers is just that, a list of numbers in order. It can be a short list, such as the number of points earned on each assignment in a class, such as {10, 10, 8, 9, 10, 6, 10}. Or it can be a longer list, even infinitely long, such as the list of prime numbers. For example, here’s a sequence of numbers, specifically, the squares of the first 12 natural numbers.

{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}

Each value in the sequence is called a term. Terms in the list are often referred to by their location in the sequence, as in the $n$ th term. For the sequence {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}, the first term of the sequence is 1, the fourth term is 16, and so on. In the sequence of assignment scores {10, 10, 8, 9, 10, 6, 10}, the first term is 10 and the third term is 8 (Figure 3.47).

A sequence of numbers. The numbers are as follows: 10, 10, 8, 9, 10, 6, and 10. A callout to the first 10 reads, First term, a sub 1. A callout to the second 10 reads, Second term, a sub 2. A callout to the third 10 reads, Fifth term, a sub 5. — **Figure 3.47** Sequence showing first, second, and fifth terms

The notation we use with sequences is a letter, which represents a term in the sequence, and a subscript, which indicates what place the term is in the sequence. For the sequence {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}, we will use the letter $a$ as a value in the sequence, and so $a sub 5$ would be the term in the sequence at the fifth position. That number is 25, so we can write $a sub 5 equals 25$ .

In this section, we focus on a special kind of sequence, one referred to as an arithmetic sequence. Arithmetic sequences have terms that increase by a fixed number or decrease by a fixed number, called the constant difference (denoted by $d$ ), provided that value is not 0. This means the next term is always the previous term plus or minus a specified, constant value. Another way to say this is that the difference between any consecutive terms of the sequence is always the same value.

To see a constant difference, look at the following sequence: {7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87}. Figure 3.48 illustrates that each term of the sequence is the previous term plus 8. Eight is the constant difference here.

A sequence of numbers. The numbers are as follows: 7, 15, 23, 31, 39, 47, 55, 63, 71, 79, and 87. Hops labeled plus 8 from each number points to the next number from left to right. — **Figure 3.48** Sequence of numbers with 8 added to each term

Example 1 Identifying Arithmetic Sequences

Determine if the following sequences are arithmetic sequences. Explain your reasoning.

${ 4, 7, 10, 13, 16, 19, 22, 25,... }$
${ 20, 40, 80, 160, 320, 640 }$
${7, 1, - 5, - 11, - 17, - 23, - 29, - 34, - 40}$

In the sequence ${ 4, 7, 10, 13, 16, 19, 22, 25,... }$ , every term is the previous term plus 3. The ellipsis indicates that the pattern continues, which means keep adding 3 to the previous term to get the new term. Therefore, this is an infinite arithmetic sequence.
In the sequence ${ 20, 40, 80, 160, 320, 640 }$ , terms increase by various amounts, for instance from term 1 to term 2, the sequence increases by 20, but from term 2 to term 3 the sequence increases by 40. So, this is not an arithmetic sequence.
In the sequence ${7, 1, - 5, - 11, - 17, - 23, - 29, - 34, - 40}$ , every term is the previous term minus 6, so this is an arithmetic sequence.

Arithmetic sequences can be expressed with a formula. When we know the first term of an arithmetic sequence, which we label $a sub 1$ , and we know the constant difference, which is denoted $d$ , we can find any other term of the arithmetic sequence. The formula for the $i th$ term of an arithmetic sequence is $a_{i} = a_{1} + d \times (i - 1)$ .

Let’s examine the formula with this arithmetic sequence: ${ 4, 7, 10, 13, 16, 19, 22, 25,... }$ . In this sequence $a sub 1 equals 4$ and $d equals 3$ . The table below shows the values calculated.

$i$ , Place in Sequence	$a_{i}$ , $i^{t h}$ Term	Value of Term	Term Written as $a_{1} + 3 \times (i - 1)$
1	$a sub 1$	4	$4 + 3 \times 0$
2	$a sub 2$	7	$4 + 3 \times 1$
3	$a sub 3$	10	$4 + 3 \times 2$
4	$a sub 4$	13	$4 + 3 \times 3$
5	$a sub 5$	16	$4 + 3 \times 4$
$i$	$a_{i}$		$4 + 3 \times (i - 1)$

We can see how the $i th$ term can be directly calculated. In this sequence, the formula is $a_{1} + 3 \times (i - 1)$ where the first term, $a sub 1$ , is 4 and the constant difference $d$ is 3. We can then determine the $47 th$ term of this sequence: $a_{47} = 4 + 3 \times (47 - 1) = 4 + 3 \times 46 = 4 + 138 = 142$ .

If we know two terms of the sequence, it is possible to determine the general form of an arithmetic sequence, $a_{i} = a_{1} + d \times (i - 1)$ .

Example 3 Determining First Term and Constant Difference Using Two Terms

A sequence is known to be arithmetic. Two of its terms are $a sub 7 equals 56$ and $a sub 19 equals 104$ . Use that information to find the constant difference, the first term, and then the $50 th$ term of the sequence.

To find the constant difference, use $d = \frac{a_{j} - a_{i}}{j - i}$ . The location of the terms is given by the subscript of the two $a$ terms, $i equals 7$ and $j equals 19$ . So, the constant difference can be calculated as such:

$d = \frac{104 - 56}{19 - 7} = \frac{48}{12} = 4$

The constant difference of 4 is then used to find $a sub 1$ .

$a_{1} = a_{i} - d (i - 1) = a_{7} - 4 (7 - 1) = 56 - 4 \times 6 = 32$

So $d equals 4$ and $a sub 1 equals 32$ .

With this information, the $50 th$ term can be found.

$a_{50} = a_{1} + d \times (i - 1) = 32 + 4 \times (50 - 1) = 32 + 4 \times 49 = 32 + 196 = 228$

The $50 th$ term is $a sub 50 equals 228$ .

Finding the Sum of a Finite Arithmetic Sequence

Sometimes we want to determine the sum of the numbers of a finite arithmetic sequence. The formula for this is fairly straightforward.

Using Arithmetic Sequences to Solve Real-World Applications

Applications of arithmetic sequences occur any time some quantity increases by a fixed amount at each step. For instance, suppose someone practices chess each week and increases the amount of time they study each week. The first week the person practices for 3 hours, and vows to practice 30 more minutes each week. Since the amount of time practicing increases by a fixed number each week, this would qualify as an arithmetic sequence.

Example 5 Applying an Arithmetic Sequence

Jordan has just watched The Queen’s Gambit and decided to hone their skills in chess. To really improve at the game, Jordan decides to practice for 3 hours the first week, and increase their time spent practicing by 30 minutes each week. How many hours will Jordan practice chess in week 20?

Jordan’s practice scheme is an arithmetic sequence, as it increases by a fixed amount each week. The first week there are 3 hours of practice. This means $a sub 1 equals 3$ . Jordan increases the time spent practicing by 30 minutes, or half an hour, each week. This means $d equals 0.5$ . Using those values, and that we want to know the amount of time Jordan will study in week 20, we determine the time in week 20 using $a_{i} = a_{1} + d \times (i - 1)$ .

$a_{20} = 3 + 0.5 \times (20 - 1) = 3 + 0.5 \times 19 = 3 + 9.5 = 12.5$

So, Jordan will practice 12.5 hours in week 20.

Example 6 Finding the Sum of a Finite Arithmetic Sequence

Let’s check back in on Jordan. Recall, Jordan had just watched The Queen’s Gambit and decided to hone their skills, practicing for 3 hours the first week, and increasing the time spent practicing by 30 minutes each week. How many hours total will Jordan have practiced chess after 30 weeks of practice?

To calculate the total amount of time that Jordan practiced, we need to use $s_{n} = n (\frac{a_{1} + a_{n}}{2})$ . The formula requires the first and last terms of the sequence. Since Jordan practiced 3 hours in the first week, the first term is $a sub 1 equals 3$ . Because we want the total practice time after 30 weeks, we need the $30 th$ term. Because the constant difference is $d equals 0.5$ , the $30 th$ term is $a_{30} = 3 + 0.5 (30 - 1) = 3 + 0.5 \times 29 = 3 + 14.5 = 17.5$ .

Applying the formula $s_{n} = n (\frac{a_{1} + a_{n}}{2})$ provides the sum of the first 30 terms.

$s_{30} = 30 (\frac{3 + 17.5}{2}) = 60 \times \frac{20.5}{2} = 615$

This means that Jordan practiced a total of 615 hours after 30 weeks.

Key Terms

sequence
term of a sequence
arithmetic sequence
first term
constant difference

Key Concepts

A sequence is a list of numbers. Any individual number in that list, or sequence, is a term of the sequence. A specific term of a sequence is denoted by the sequence symbol with a subscript indicating where the term in the sequence is.
A special form of a sequence is an arithmetic sequence. Each arithmetic sequence is determined by its first term and its constant difference. Any term in an arithmetic sequence is determined by adding the constant difference to the preceding term.
If the first term and the constant difference of an arithmetic sequence are known, then any term of the sequence can be found directly.
Because arithmetic sequences follow such a strict pattern, the sum of the first $n$ terms of an arithmetic sequence can be determined with the formula $s_{n} = n (\frac{a_{1} + a_{n}}{2})$ .

Formulas

$a_{i} = a_{1} + d \times (i - 1)$
$d = \frac{a_{j} - a_{i}}{j - i}$
$a_{1} = a_{i} - d (i - 1)$
$s_{n} = n (\frac{a_{1} + a_{n}}{2})$

Videos

Adapted from Contemporary Mathematics by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.