Arithmetic Sequences

Guide to Arithmetic
Sequences

Drag and Drop

Can you predict the pattern?



3, 9, 15, 21, 27, 33, ​ ​

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39

45

51

36

44

50

40

46

52

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What about this one? 12, 24, 36, 48, 60, 72​

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84

96

108

82

86

94

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100

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106

Cue Math defines as “the sequence where the common difference remains constant between any two successive terms.” 

https://www.cuemath.com/algebra/arithmetic-sequence/

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​Arithmetic Sequences

​Basically it is a group of numbers which contain a pattern with the same difference between a number and the one next to it.

• Can start with any first term

• Must increase or decrease by a constant amount between each term in the sequence.

• Cannot skip any terms

​

​Characteristics of an arithmetic sequence:

​Look for the same amount being added or subtracted with each new term.

Multiple Select

Which of the following would be an arithmetic sequence?

• 1, 3, 6, 10, 15
• 5, 10, 15, 20, 25
• 3, 6, 9, 12, 15
• 1, 2, 4, 6, 10, 16

If you were to take $25 out of your paycheck each week and put it in a savings account, the growth of the savings account would be an arithmetic sequence.

In Banking: regular deposits of same amount

​When starting an exercise plan it is recommended that you start slowly, adding to your plan at a gradual pace. This can be done as an arithmetic sequence, add 2 new sets each week, until they hit their desired set total. So, if a person starts with 3 sets, he can move to 5 sets the next week, 7 sets the third week, and so on.

In Health: An increasing exercise plan

​Where we might see an arithmetic sequence: Daily Life

With pyramid like patterns in stadiums and auditoriums where seats are designed around a center location, the amount of seats in each row are often in an arithmetic sequences. For example, a seat might be added to each side of a row as the rows get further from the center point, this would be an arithmetic sequence of +2 each row.

In Structures: Building seats in auditoriums and stadiums

“For example, if a company purchased a truck for $35,000 and it depreciates at the constant rate of $700 per month, its value for each month will follow an arithmetic sequence.” (The Boffins Portal Team. (2022, January 5).

In Accounting: Calculating depreciating assets

“A company may embark on a production expansion over a period of time. For example, it can increase its production by 20 units each week.  If its production in the first week was 200, its production for the first 4 weeks is 200, 220, 240, 260…which follows an arithmetic sequence.” (The Boffins Portal Team. (2022, January 5).

In Factories: Production Plans

Where we might see an arithmetic sequence: In Companies

There is a formula for that!

Lets look at the sequence 7, 12, 17, 22, 27
a₁ = 7
a₂= 7+5
a₃= 7+5+5
a₄= 7+5+5
a₅= 7+5+5+5+5

We can see that the number of 5s added to the 7 is one less than the term's number in the sequence.
This is what gives us the formula to the left.

What if we want to find a future term but not all the terms in between?

We can see that that a₁ is 6 which is the first term.

Next we need to figure out the difference between each term. We can solve -30-(-12) = -18 and -48-(-30) = -18

So we can input that into the formula so that
a_n = a₁ + d (n-1)

a_n = 6 + -18 (n-1)

Write the general term of each arithmetic sequence 6, –12, –30, –48, –66, ...

We can see that that a₁ is 13 which is the first term.

Next we need to figure out the difference between each term. We can solve 22-13 = 9 and 31-22 = 9

So we can input that into the formula so that
a_n = a₁ + d (n-1)

a_n = 13 + 9 (n-1)

Write the general term of each arithmetic sequence 13, 22, 31, 40, 49, ...

Let's look at some examples

First we need to find out general term formula:

a₁ = 45 and the difference is -6

So we can input that into the formula so that
a_n = 45 + -6 (n-1)

Then we can input the 30 in for the nth term.
a₃₀ = 45 + -6 (30-1) => 45 + -174 = -129

Find the 30th term of the sequence 45, 39, 33, 27, 21, ...

First we need to find out general term formula:

a₁ = -121 and the difference is 34

So we can input that into the formula so that
a_n = -121 + 34 (n-1)

Then we can input the 27 in for the nth term.
a₂₇ = -121 + 34 (27-1) => -121 + 884 = 763

Calculate the 27th term in the arithmetic progression –121, –87, –53, –19, 15, ...

How about a few more?

Now it is time for you to try....

Multiple Choice

Write the general term of each arithmetic sequence –199, –99, 1, 101, 201, ...

• a_n = -199 + -100 (n-1)

• a_n = -199 + 100 (n-1)

• a_n = 100 + -199 (n-1)

• a_n = -199 + 100 (n+1)

Write the general term of each arithmetic sequence –199, –99, 1, 101, 201, ...

We can see that that a₁ is -199 which is the first term.

Next we need to figure out the difference between each term. We can solve -99-(-199) = 100 and 1-(-99) = 100

So we can input that into the formula so that
a_n = a₁ + d (n-1)

a_n = -199 + 100 (n-1)

Multiple Choice

Write the general term of each arithmetic sequence 57, 52, 47, 42, 37, ...

• a_n = 57 + 5 (n-1)

• a_n = -5 + 57 (n-1)

• a_n = 57 + -5 (n+1)

• a_n = 57 + -5 (n-1)

Write the general term of each arithmetic sequence 57, 52, 47, 42, 37, ...

We can see that that a₁ is 57 which is the first term.

Next we need to figure out the difference between each term. We can solve 52- 57 = -5 and
52- 47 = -5

So we can input that into the formula so that
a_n = a₁ + d (n-1)

a_n = 57 + -5 (n-1)

​Calculate the 47th term in the arithmetic progression 25, 16, 7, –2, –11, ...

First we need to find out general term formula:

a₁ = 25 and the difference is -9

So we can input that into the formula so that
a_n = 25 + -9 (n-1)

Then we can input the 47 in for the nth term.
a₄₇ = 25 + -9 (47-1) => 25 + -414 = -389

Given the arithmetic progression 131, 159, 187, 215, 243, ... find the 22nd term.

First we need to find out general term formula:

a₁ = 131 and the difference is 28

So we can input that into the formula so that
a_n = 131 + 28 (n-1)

Then we can input the 22 in for the nth term.
a₂₂ = 131 + 28 (22-1) => 131 + 588 = 719

References

Cuemath. (n.d.). Arithmetic sequence - formula, definition, examples, applications: Arithmetic series.  https://www.cuemath.com/algebra/arithmetic-sequence/ 

Math is Fun (n.d.).  Arithmetic Sequences and Sums. Arithmetic sequences and sums. https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html 

Nerdstudy. (2017, February 12). Arithmetic sequences - nerdstudy. YouTube. https://www.youtube.com/watch?v=eNJp6Nozqlw 

Printable Worksheets for Kids. (n.d.).  Arithmetic sequence worksheets. https://www.mathworksheets4kids.com/arithmetic-sequence.php 

The Boffins Portal Team. (2022, January 5). 10 examples of arithmetic sequences in real life. The Boffins Portal. https://boffinsportal.com/10-examples-of-arithmetic-sequences-in-real-life/