Arithmetic Sequences

Learning Objectives

After completing this section, you should be able to:

1. Identify arithmetic sequences.
2. Find a given term in an arithmetic sequence.
3. Find the \(n\)th term of an arithmetic sequence.
4. Find the sum of a finite arithmetic sequence.
5. Use arithmetic sequences to solve real- world applications

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 ().

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 \({a}_{5}\) would be the term in the sequence at the fifth position. That number is 25, so we can write \({a}_{5}=25\).

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}. illustrates that each term of the sequence is the previous term plus 8. Eight is the constant difference here.

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

Condensed — the full section is in OpenStax Contemporary Mathematics.

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.

Finding the Sum of a Finite Arithmetic Sequence

Try it.

What is the sum of the first 60 terms of an arithmetic sequence with \({a}_{1}=4.5\) and \(d=2.5\)?

Solution

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

\[{a}_{60}=4.5+2.5(60-1)=4.5+2.5\times 59=4.5+147.5=152\]

Applying the formula \({s}_{n}=n(\frac{{a}_{1}+{a}_{n}}{2})\) provides the sum of the first 60 terms.

\[{s}_{60}=60(\frac{4.5+152}{2})=60\times \frac{156.5}{2}=4,695\]

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

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.

Applying an Arithmetic Sequence

Try it.

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?

Solution

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 \({a}_{1}=3\). 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.

Finding the Sum of a Finite Arithmetic Sequence

Try it.

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?

Solution

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 \({a}_{1}=3\). Because we want the total practice time after 30 weeks, we need the \(30\text{th}\) term. Because the constant difference is \(d=0.5\), the \(30\text{th}\) 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.

Condensed — the full section is in OpenStax Contemporary Mathematics.

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 \({s}_{n}=n(\frac{{a}_{1}+{a}_{n}}{2})\).

Videos

• Arithmetic Sequences
• Finding the First Term and Constant Difference for an Arithmetic Sequence
• Finding the Sum of a Finite Arithmetic Sequence
• Fibonacci Sequence and “Lateralus”

Practice (6)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

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

   1. \(\{4,7,10,13,16,19,22,25,...\}\)
   2. \(\{20,40,80,160,320,640\}\)
   3. \(\{7,1,-5,-11,-17,-23,-29,-34,-40\}\)
   जवाब दिखाएँ

   1. 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.
   2. 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.
   3. 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.

2. Identify \({a}_{1}\) and \(d\) for the following arithmetic sequence. Use this information to determine the \(60\text{th}\) term. \[\{18,31,44,57,70,83,...\}\]
   

   जवाब दिखाएँ

   Inspecting the sequence shows that \({a}_{1}=18\) and \(d=13\). We use those values in the formula, with \(i=60\).

   \[{a}_{60}={a}_{1}+d\times (i-1)=18+13\times (60-1)=18+13\times 59=18+767=785\]

3. A sequence is known to be arithmetic. Two of its terms are \({a}_{7}=56\) and \({a}_{19}=104\). Use that information to find the constant difference, the first term, and then the \(50\text{th}\) term of the sequence.

   जवाब दिखाएँ

   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, \(i=7\) and \(j=19\). 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 \({a}_{1}\).

   \[{a}_{1}={a}_{i}-d(i-1)={a}_{7}-4(7-1)=56-4\times 6=32\]

   So \(d=4\) and \({a}_{1}=32\).

   With this information, the \(50\text{th}\) 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 \(50\text{th}\) term is \({a}_{50}=228\).

4. What is the sum of the first 60 terms of an arithmetic sequence with \({a}_{1}=4.5\) and \(d=2.5\)?

   जवाब दिखाएँ

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

   \[{a}_{60}=4.5+2.5(60-1)=4.5+2.5\times 59=4.5+147.5=152\]

   Applying the formula \({s}_{n}=n(\frac{{a}_{1}+{a}_{n}}{2})\) provides the sum of the first 60 terms.

   \[{s}_{60}=60(\frac{4.5+152}{2})=60\times \frac{156.5}{2}=4,695\]

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

5. 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?

   जवाब दिखाएँ

   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 \({a}_{1}=3\). 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.

6. 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?

   जवाब दिखाएँ

   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 \({a}_{1}=3\). Because we want the total practice time after 30 weeks, we need the \(30\text{th}\) term. Because the constant difference is \(d=0.5\), the \(30\text{th}\) 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.

Symbols used here

i

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

\prod_{k=1}^{n} a_k

product
Multiply a_k for k = 1 up to n.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

\varphi(n),\ \pi(x)

Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

a \bmod n

remainder
What is left after dividing a by n.

How to: Arithmetic Sequences

1. Identify arithmetic sequences.
2. Find a given term in an arithmetic sequence.
3. Find the
4. Find the sum of a finite arithmetic sequence.
5. Use arithmetic sequences to solve real- world applications
6. In the sequence
7. In the sequence
8. In the sequence

Questions people ask

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller–Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

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Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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