3.10: Arithmetic Sequences

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Learning Objectives

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 $n$

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 $\PageIndex{1}$: 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 a5a5 would be the term in the sequence at the fifth position. That number is 25, so we can write $a_{5} =$ 25a5=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 3.49 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 $\PageIndex{2}$: Sequence of numbers with 8 added to each term

Example $\PageIndex{1}$ : Identifying Arithmetic Sequences

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

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

Answer

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.

Your Turn $\PageIndex{2}$

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

$\left\{ {7.6,5.4,3.2,1.0, - 1.2, - 3.4, - 5.6, - 7.8, - 10.0} \right\}$

$\left\{ {14,16,18,22,28,40,32,0} \right\}$

$\left\{ {14,20,26,32,38,44,50,56,62,68,74,80,...} \right\}$

Arithmetic sequences can be expressed with a formula. When we know the first term of an arithmetic sequence, which we label a1a1, and we know the constant difference, which is denoted $d$ , we can find any other term of the arithmetic sequence. The formula for the ithith term of an arithmetic sequence is $a_{i} = a_{1} + d \times (i - 1)$ .

FORMULA

If we have an arithmetic sequence with first term a1a1 and constant difference $d$ , then the ithith term of the 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 $a_{1} =$ 4a1=4 and $d = 3$ . The table below shows the values calculated.

$i$ , Place in Sequence	aiai, ithith Term	Value of Term	Term Written as $a_{1} + 3 \times (i - 1)$
1	a1a1	4	$4 + 3 \times 0$
2	a2a2	7	$4 + 3 \times 1$
3	a3a3	10	$4 + 3 \times 2$
4	a4a4	13	$4 + 3 \times 3$
5	a5a5	16	$4 + 3 \times 4$
$i$	aiai		$4 + 3 \times (i - 1)$

We can see how the ithith term can be directly calculated. In this sequence, the formula is $a_{1} + 3 \times (i - 1)$ where the first term, a1a1, is 4 and the constant difference $d$ is 3. We can then determine the 47th47th term of this sequence: a47=4+3×(47−1)=4+3×46=4+138=142a47=4+3×(47−1)=4+3×46=4+138=142.

Example $\PageIndex{3}$: Calculating a Term in an Arithmetic Sequence

Identify a1a1 and $d$ for the following arithmetic sequence. Use this information to determine the 60th60th term.

${18, 31, 44, 57, 70, 83, ...}$

Answer: Inspecting the sequence shows that $a_{1} =$ 18a1=18 and $d =$ 13d=13. We use those values in the formula, with $i =$ 60i=60.
a60=a1+d×(i−1)=18+13×(60−1)=18+13×59=18+767=785a60=a1+d×(i−1)=18+13×(60−1)=18+13×59=18+767=785

Your Turn $\PageIndex{3}$

Identify ${a_1}$ and $d$ for the following arithmetic sequence. Use this information to determine the $86\text{th}$ term.

$\left\{ {4.5,8.1,11.7,15.3,18.9,22.5,26.1,...} \right\}$

Video

Arithmetic Sequences

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

FORMULA

If we have the $i$ th term of an arithmetic sequence, aiai, and the $j$ th term of the sequence, ajaj, then the constant difference is $d = a_{j} -$ aij−id=aj−aij−i and the first term of the sequence is $a_{1} = a_{i} - d (i - 1)$ .

Example $\PageIndex{4}$: Determining First Term and Constant Difference Using Two Terms

A sequence is known to be arithmetic. Two of its terms are $a_{7} =$ 56a7=56 and a19=104a19=104. Use that information to find the constant difference, the first term, and then the 50th50th term of the sequence.

Answer

To find the constant difference, use $d = a_{j} -$ aij−id=aj−aij−i. The location of the terms is given by the subscript of the two $a$ terms, $i = 7$ and $j =$ 19j=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 a1a1.

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

=ai−d(i−1)=a7−4(7−1)=56−4×6=32.

So $d = 4$ and $a_{1} =$ 32a1=32.

With this information, the 50th50th term can be found.

a50=a1+d×(i−1)=32+4×(50−1)=32+4×49=32+196=228

a50

=a1+d×(i−1)=32+4×(50−1)=32+4×49=32+196=228.

The 50th50th term is a50=228a50=228.

Your Turn $\PageIndex{4}$

A sequence is known to be arithmetic. Two of the terms are ${a_{14}} = 41$ and ${a_{38}} = 161$. Use that information to find the constant difference and the first term. Then determine the ${151\text{st}}$ term of the sequence.

Video

Finding the First Term and Constant Difference for an Arithmetic Sequence

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.

FORMULA

The sum of the first $n$ terms of a finite arithmetic sequence, written snsn, with first and last term a1a1 and anan, respectively, is $s_{n} = n (a_{1} +$ an2)sn=n(a1+an2).

Example $\PageIndex{5}$ : Finding the Sum of a Finite Arithmetic Sequence

What is the sum of the first 60 terms of an arithmetic sequence with $a_{1} = 4.5$ and $d = 2.5$ ?

Answer

The formula requires the first and last terms of the sequence. The first term is given, $a_{1} = 4.5$ . The 60th60th term is needed. Using the formula $a_{1} = a_{i} + d (i - 1)$ provides the value for the 60th60th term.

a60=4.5+2.5(60−1)=4.5+2.5×59=4.5+147.5=152

a60

=4.5+2.5(60−1)=4.5+2.5×59=4.5+147.5=152.

Applying the formula $s_{n} = n (a_{1} +$ an2)sn=n(a1+an2) provides the sum of the first 60 terms.

s60=60(4.5+1522)=60×156.52=4,695

s60

=60(4.5+1522)=60×156.52=4,695.

The sum of the first 60 terms is 4,695.

Your Turn $\PageIndex{5}$

What is the sum of the first 101 terms of an arithmetic sequence with ${a_1} = 13$ and $d = 2.25$?

Video

Finding the Sum of a Finite Arithmetic Sequence

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 $\PageIndex{6}$: 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?

Answer

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 $a_{1} =$ 3a1=3. Jordan increases the time spent practicing by 30 minutes, or half an hour, each week. This means $d = 0.5$ . 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.

Your Turn $\PageIndex{6}$

Christina decides to save money for after graduation. Christina starts by setting aside $10. Each week, Christina increases the amount she saves by $5. How much money will Christina save in week 52?

Example $\PageIndex{7}$: 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?

Answer

To calculate the total amount of time that Jordan practiced, we need to use $s_{n} = n (a_{1} +$ an2)sn=n(a1+an2). The formula requires the first and last terms of the sequence. Since Jordan practiced 3 hours in the first week, the first term is $a_{1} =$ 3a1=3. Because we want the total practice time after 30 weeks, we need the 30th30th term. Because the constant difference is $d = 0.5$ , the 30th30th term is a30=3+0.5(30−1)=3+0.5×29=3+14.5=17.5a30=3+0.5(30−1)=3+0.5×29=3+14.5=17.5.

Applying the formula $s_{n} = n (a_{1} +$ an2)sn=n(a1+an2) provides the sum of the first 30 terms.

s30=30(3+17.52)=60×20.52=615

s30

=30(3+17.52)=60×20.52=615.

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

Your Turn $\PageIndex{7}$

In a theater, the first row has 24 seats. Each row after that has 2 more seats. How many total seats are there if there are 40 rows of seat in the theater?

Who Knew?: The Fibonacci Sequence

Not all sequences are arithmetic. One special sequence is the Fibonacci sequence, which is the sequence that has as its first two terms 1 and 1. Every term thereafter is the sum of the previous two terms. The first nine terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, and 34.

This sequence is found in nature, architecture, and even music! In nature, the Fibonacci sequence describes the spirals of sunflower seeds, certain galaxy spirals, and flower petals. In music, the band Tool used the Fibonacci sequence in the song “Lateralus.” The Fibonacci sequence even relates to architecture, as it is closely related to the golden ratio.

Video

Fibonacci Sequence and “Lateralus”

Check Your Understanding

1. Is the following an arithmetic sequence? Explain.

{3, 6, 9, 15, 25, 39, 90}

2. What is the 7th term of the following sequence?

{1, 5, 7, 100, 4, -17, 8, 100, 19, 7.6, 345}

3. In an arithmetic sequence, the first term is 10 and the constant difference is 4.5. What is the 135th term?

4. If the eighth term of an arithmetic sequence is 35 and the 40th term is 131, what is the constant difference and the first term of the sequence?

5. What is the sum of the first 100 terms of the arithmetic sequence with first term 4 and constant difference 7?

6. A new marketing firm began with 30 people in its survey group. The firm adds 4 people per day. How many people will be in their survey group after 100 days?

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Example \(\PageIndex{1}\) : Identifying Arithmetic Sequences

Your Turn \(\PageIndex{2}\)

FORMULA

Example \(\PageIndex{3}\): Calculating a Term in an Arithmetic Sequence

Your Turn \(\PageIndex{3}\)

Video

FORMULA

Example \(\PageIndex{4}\): Determining First Term and Constant Difference Using Two Terms

Your Turn \(\PageIndex{4}\)

Video

FORMULA

Example \(\PageIndex{5}\) : Finding the Sum of a Finite Arithmetic Sequence

Your Turn \(\PageIndex{5}\)

Video

Example \(\PageIndex{6}\): Applying an Arithmetic Sequence

Your Turn \(\PageIndex{6}\)

Example \(\PageIndex{7}\): Finding the Sum of a Finite Arithmetic Sequence

Your Turn \(\PageIndex{7}\)

Who Knew?: The Fibonacci Sequence

Video