Not sure where to start? These are the lessons and games we recommend most to students, parents and teachers on Basic-mathematics.com. Each lesson explains the topic in plain language, walks you through worked examples step by step, and ends with practice so you can check your understanding. Pick one below, or use the menu to find lessons on fractions, algebra, geometry, statistics and more.
An arithmetic sequence is a sequence where each term is found by adding or subtracting the same value from one term to the next. This value that is added or subtracted is called the "common difference".
If the common difference is positive, the terms of the sequence will increase in value.
If the common difference is negative, the terms of the sequence will decrease in value.
For example, the following two sequences are examples of arithmetic sequences.
1, 4, 7, 10, 13, 16, 19, ...
As you can see, each term is found by adding 3, the common difference, to the previous term.
70, 62, 54, 46, 38, ...
This time, to find each term, we subtract 8 from the previous term. The common difference is −8.
What is the common difference of 15, 11, 7, 3, ...?
−4. Each term is 4 less than the one before: 11 − 15 = −4.
Many arithmetic sequences can be modeled with an algebraic expression
Here is a trick or "recipe per se" to quickly get an algebraic expression!
Expression = (the number added each time) × n + (the number that comes right before the first term).
1) Let us try to model 1, 4, 7, 10, 13, 16, 19, ...
Let n represent any term number in the sequence. The number we add to each term is 3.
The number that comes right before 1 in the sequence is −2.
We can therefore model the sequence with this algebra expression: 3 × n + (−2).
Check to see if the algebra expression works:
When n = 1, which represents the first term, we get 3 × 1 + (−2) = 3 + (−2) = 1
When n = 2, which represents the second term, we get 3 × 2 + (−2) = 6 + (−2) = 4
When n = 3, which represents the third term, we get 3 × 3 + (−2) = 9 + (−2) = 7
The algebraic expression works!
2) Let us try to model 70, 62, 54, 46, 38, ...
Let n represent any term number in the sequence. The number we add to each term is −8.
The number that comes right before 70 in the sequence is 78.
We can therefore model the sequence with this algebraic expression: −8 × n + 78.
Check to see if the algebra expression works:
When n = 1, which represents the first term, we get −8 × 1 + 78 = −8 + 78 = 70
When n = 2, which represents the second term, we get −8 × 2 + 78 = −16 + 78 = 62
When n = 3, which represents the third term, we get −8 × 3 + 78 = −24 + 78 = 54
Again, the algebraic expression works!
Write an algebraic expression for 2, 7, 12, 17, ...
5 × n + (−3). The number added each time is 5, and the number right before 2 is 2 − 5 = −3. Check: n = 1 gives 5 − 3 = 2.
The way that we modeled the arithmetic sequences above with algebraic expressions is a shortcut. We will now look for the arithmetic sequence formula using the algebraic expressions.
1) 3 × n + (−2) is the algebraic expression for 1, 4, 7, 10, 13, 16, 19, ...
Let us try to rewrite 3 × n + (−2) by making the first term appear in the expression.
3 × n + (−2) = 3 × n + (−3) + 1 (since −2 = −3 + 1)
3 × n + (−2) = 3 × (n − 1) + 1
3 is the number we add to each term
1 is the first term
n is the term number
2) −8 × n + 78 is the algebraic expression for 70, 62, 54, 46, 38, ...
Let us try to rewrite −8 × n + 78 by making the first term appear in the expression.
−8 × n + 78 = −8 × n + 8 + 70 (since 78 = 8 + 70)
−8 × n + 78 = −8 × (n − 1) + 70
−8 is the number we add to each term
70 is the first term
n is the term number
In general,
Let d be the number we add each time or the common difference.
Let a1 be the first term
Let n be the term number
Let an be the nth term.
Then,
Writing d × n instead of d × (n − 1). To get from the first term to the nth term, you add d only n − 1 times. For 1, 4, 7, ..., the 3rd term is 3 × (3 − 1) + 1 = 7. Using 3 × 3 + 1 = 10 gives the 4th term instead.
Answer: a20 = 81
Find the 10th term of 3, 5, 7, 9, ...
21. d = 2 and a1 = 3, so a10 = 2 × (10 − 1) + 3 = 18 + 3 = 21.
Are the given sequences arithmetic? If so, find the 98th term.
a. 2, 6, 9, 11, ...
2, 6, 9, 11, ... is not an arithmetic sequence since the number we add to each term is not always the same.
b. −4, 0, 4, 8, 12, ...
−4, 0, 4, 8, 12, ... is an arithmetic sequence since the number we add to each term is always the same.
an = d × (n − 1) + a1
d = 4
n = 98
a1 = −4
a98 = 4 × (98 − 1) + a1
a98 = 4 × (97) + (−4)
a98 = 388 + (−4)
a98 = 384
Is 1, 3, 6, 10, ... an arithmetic sequence?
No. The differences are 2, 3 and 4, so the number added is not always the same.
Now check what you learned. Every answer is explained, and each time you play you get new questions.
Quiz Complete!