What is an Arithmetic Sequence? (Video & Practice Questions)

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Hey guys! Welcome to this video on arithmetic sequences and their formulas!

Sequences typically have set patterns that enable us to predict what each term might be. A list of numbers ordered in a particular way is a sequence, and each individual number is referred to as a term.

Understanding arithmetic sequences, and how to identify them is a great way to develop critical thinking skills. Identifying how numbers relate to each other, and what commonalities they have in a sequence can carry over into better critical thinking skills in other intellectual endeavors.

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence or progression of numbers where the difference between each number is the same (or constant).

For example, in the series 5, 12, 19, 26… , we can tell that this is an arithmetic sequence by subtracting each number from the one following it.

Sequence:

5 <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>

12 <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math>

19 <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>19</mn></math>

26 <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>26</mn></math>

5 - 12 = - 7 <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>-</mo><mn>12</mn><mo>=</mo><mo>-</mo><mn>7</mn></math>

12 - 19 = - 7 <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>-</mo><mn>19</mn><mo>=</mo><mo>-</mo><mn>7</mn></math>

19 - 26 = - 7 <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>19</mn><mo>-</mo><mn>26</mn><mo>=</mo><mo>-</mo><mn>7</mn></math>

The difference is the same each time, therefore it is a constant.

Using Sequences

In other words, to tell whether or not it is an arithmetic sequence, we need to be able to see what is happening between each number, and whatever happens between two needs to be the same thing that is happening between each consecutive number.

Another way to write this arithmetic sequence is:

(a, a + d, a + 2 d, a + 3 d, \dots) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo>,</mo><mi>a</mi><mo>+</mo><mn>2</mn><mi>d</mi><mo>,</mo><mi>a</mi><mo>+</mo><mn>3</mn><mi>d</mi><mo>,</mo><mo>\dots</mo><mo stretchy="false">)</mo></math>

The letter $a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>$ represents the first term, and the letter $d <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>$ represents the constant difference.

Example #1

So, looking back at our sequence of numbers let’s apply this. Using this method, let’s plug in our numbers.

We have ‘5’ as our first term, and we know that the constant difference is 7.

5, 5 + 7, 5 + 2 (7), 5 + 3 (7) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>,</mo><mn>5</mn><mo>+</mo><mn>7</mn><mo>,</mo><mn>5</mn><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo>,</mo><mn>5</mn><mo>+</mo><mn>3</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></math>

= 5, 12, 19, 26 <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>5</mn><mo>,</mo><mn>12</mn><mo>,</mo><mn>19</mn><mo>,</mo><mn>26</mn></math>

What if we wanted to find the 50th term of this sequence?

Let’s write an arithmetic sequence as a formula:

x n = a + d (n - 1) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo></math>

The reason $n - 1 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mrow data-mjx-texclass="ORD"><mo>-</mo></mrow><mn>1</mn></math>$ is used is because $d <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>$ , the constant difference, is not applied to the very first term.

Example #2

Let’s take a look at a new arithmetic sequence, and, using our new formula, calculate a certain term.

Look at this sequence:

9, 17, 25, 33, 41, 49, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>,</mo><mn>17</mn><mo>,</mo><mn>25</mn><mo>,</mo><mn>33</mn><mo>,</mo><mn>41</mn><mo>,</mo><mn>49</mn><mo>,</mo></math>

57, 65, 73, 81, \dots <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>57</mn><mo>,</mo><mn>65</mn><mo>,</mo><mn>73</mn><mo>,</mo><mn>81</mn><mo>,</mo><mo>\dots</mo></math>

So, we are going to calculate the fourth term using our formula. We know that our first term is 9, so now we have to calculate the constant difference, which is 8.

So we are looking for our fourth term which is represented by $x 4 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>4</mn></msub></math>$ . $a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>$ is our first term (which is 9), $d <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>$ is our constant difference (which is 8), and $n <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>$ represents the number’s sequence order (which in this case is 4).

So, let’s plug in our numbers.

x 4 = 9 + 8 (4 - 1) = 33 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>4</mn></mrow></msub><mo>=</mo><mn>9</mn><mo>+</mo><mn>8</mn><mo stretchy="false">(</mo><mn>4</mn><mrow data-mjx-texclass="ORD"><mo>-</mo></mrow><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>33</mn></math>

Now, typically, you wouldn’t use this formula to calculate a term that is already listed, but rather to predict and calculate a term farther along the progression of numbers.

So, let’s say we want to calculate the 1,698th term of this arithmetic sequence.

x 1, 698 = 9 + 8 (1, 698 - 1) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mn>698</mn><mo>=</mo><mn>9</mn><mo>+</mo><mn>8</mn><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>698</mn><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo></math>

= 13, 585 <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>13</mn><mo>,</mo><mn>585</mn></math>

I hope that this video was helpful!

See you next time!

Frequently Asked Questions

Q

What is arithmetic sequence?

A

An arithmetic sequence is a sequence in which the same number is added or subtracted from one term to the next.

Q

How do you find the sum of an arithmetic sequence?

A

To find the sum of an arithmetic sequence, use this formula:
$sn=n2(a1+an)<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mi>n</mi></msub><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>$
n:the position you are adding up to
$a 1 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub></math>$ :the first element of the sequence
$a n <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub></math>$ :the element in the position you are adding up to
Ex. What is the sum of the first 7 elements of the sequence $a n = 4 + (n - 1) \times 3 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>4</mn><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo><mi>\times</mi><mn>3</mn></math>$ ?
$n = 7 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>7</mn></math>$
$a 1 = 4 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>4</mn></math>$
$a n = a 7 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mn>7</mn></msub></math>$ $= 4 + (7 - 1) \times 3 = 4 + 6 \times 3 <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>+</mo><mo stretchy="false">(</mo><mn>7</mn><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo><mi>\times</mi><mn>3</mn><mo>=</mo><mn>4</mn><mo>+</mo><mn>6</mn><mi>\times</mi><mn>3</mn></math>$ $= 4 + 18 = 22 <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>+</mo><mn>18</mn><mo>=</mo><mn>22</mn></math>$
$s7=72(4+22)=72(26)=91<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mn>7</mn></msub><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mn>4</mn><mo>+</mo><mn>22</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mn>26</mn><mo stretchy="false">)</mo><mo>=</mo><mn>91</mn></math>$

Q

How do you find the nth term of an arithmetic sequence?

A

To find the n^th term of an arithmetic sequence, use this formula:
$a n = a 1 + (n - 1) d <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo><mi>d</mi></math>$
Where a_n is the term you are looking for, a_1 is the first term of the sequence, n is the position of the term you are looking for, and d is the common difference.
Ex. What is the 17th term of this sequence: 1, 3, 5, 7, . . .?
$a 1 = 1 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></math>$
$n = 17 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>17</mn></math>$
$d = 2 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>2</mn></math>$ (you add 2 to get to the next term)
$a 1 7 = 1 + (17 - 1) (2) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub><mn>7</mn><mo>=</mo><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mn>17</mn><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math>$ $= 1 + (16) (2) = 1 + 32 = 33 <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>+</mo><mn>32</mn><mo>=</mo><mn>33</mn></math>$

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Arithmetic Sequence