An arithmetic sequence is a sequence where the step from one term to the next is constant. That is, you always add the same thing to get from one term to the next.
An arithmetic sequence is like going up a huge flight of stairs. Most days, you'll walk up the flight of stairs one step at time. Sometimes, though, you are in a hurry, and you skip stairs as you rush up the stairs to get your coat before the bus comes. The increment between each of your steps is constant: one stair if you are on time, and two stairs if you are in a hurry. Be careful not to trip. A bruised shin is the worst.
Sample Problem
The sequence
2, 4, 6, 8, 10, ...
is an arithmetic sequence. To get from one term to the next, we always add 2.
Sample Problem
The sequence
2, 4, 8, 16, ...
is not an arithmetic sequence. To get from 2 to 4 we add 2, but to get from 4 to 8 we add 4.
Since we didn't add the same thing every time, this isn't arithmetic.
Saying the step up from one term to the next term is constant is the same as saying the step down from one term to the previous term is constant. In an arithmetic sequence, the difference between one term and the previous term must always be the same. Our base-running analogy breaks down here. You wouldn't want to run them backwards unless you go from first to second base facing the wrong way.
Sample Problem
Is the sequence
4, 8, 12, 16, ...
arithmetic?
Answer.
Let's find the difference between each pair of successive terms.
8 – 4 = 4
12 – 8 = 4
16 – 12 = 4
The difference between each pair of consecutive terms is 4.
That means this is an arithmetic sequence.
Sample Problem
Is the sequence
3, 6, 10, 15, 21, ...
arithmetic?
Answer.
Let's find the difference between each pair of terms.
6 – 3 = 3
10 – 6 = 4
We don't have to go any further. The difference between successive terms isn't always the same. That means this is not an arithmetic sequence.
It's also okay for the step from one term to the next to be negative, as long as the step is constant.
Sample Problem
The sequence
10, 0, -10, -20, ...
is an arithmetic sequence because the step from one term to the next is always -10.
Now that we're clear on what an arithmetic sequence is, let's put the definition into symbols and an equation. Mathematicians love equations.
In an arithmetic sequence, to get from one term to the next term we add some constant d.
In symbols,
an + 1 = an + d.
This is the same as saying that to get from one term to the previous term we subtract some constant d.
In symbols,
an + 1 – an = d.
Arithmetic sequences are usually defined in terms of subtraction rather than addition. The value d is called the common difference for the sequence.
We'll start all our arithmetic sequences with n = 1 corresponding to the first term, just because we can. We can also jump up on our desks and sing, "Take Me Out to the Ball Game," at the top of our lungs, but it probably won't help us with sequences.
An arithmetic sequence is completely determined by two things: its starting term a1 and its common difference d. Once you know where an arithmetic sequence starts and what its step size is, you know everything there is to know about it.
If we throw a baseball backwards behind us, we might break our mother's chandelier. But if we know two terms in an arithmetic sequence, we can work backwards to figure out a1 and d.
This means we need to know how many steps it takes to get from one term am to another term an.
Example 1
Write the first four terms of the arithmetic sequence with a1 = 4 and d = 5. |
Example 2
If a1 = 3 and d = 5, find a5. |
Example 3
If a1 = 12 and d = -2, find a11. |
Example 4
An arithmetic sequence has a5 = 11 and a6 = 13. Find a1 and d. |
Example 5
An arithmetic sequence has a10 = 52 and a11 = 56. Find a1 and d. |
Example 6
Find a1 and d for the arithmetic sequence with a2 = 8 and a4 = 28 |
Example 7
Find a1 and d for the arithmetic sequence with a50 = 14 and a100 = -136. |
Exercise 1
Determine whether the sequence is arithmetic or not.
1, 2, 3, 4, 5,...
Exercise 2
Determine whether the sequence is arithmetic or not.
10, 13, 16, 20,...
Exercise 3
Determine whether the sequence is arithmetic or not.
1, 3, 5, 7, 9,...
Exercise 4
Determine whether the sequence is arithmetic or not.
15, 21, 26, 30,...
Exercise 5
Determine whether the sequence is arithmetic or not.
0, 10, 20, 30,...
Exercise 6
Determine whether the sequence is arithmetic or not.
1, -1, 1, -1,...
Exercise 7
Determine whether the sequence is arithmetic or not.
5, -10, 15, -20,...
Exercise 8
Determine whether the sequence is arithmetic or not.
0, -2, -4, -6,...
Exercise 9
Write the first four terms of the arithmetic sequence with
a1 = 10 and d = 7
Exercise 10
Write the first four terms of the arithmetic sequence with
a1 = 5 and d = -2
Exercise 11
Write the first four terms of the arithmetic sequence with
a1 = -1 and d = -3
Exercise 12
How many steps does it take to get from
a1 to a6?
Exercise 13
How many steps does it take to get from
a1 to a75?
Exercise 14
How many steps does it take to get from
a1 to an?
Exercise 15
Find the requested term for the arithmetic sequence.
Find a8 if a1 = 20 and d = 3.
Exercise 16
Find the requested term for the arithmetic sequence.
Find a11 if a1 = 1 and d = -1.
Exercise 17
Find the requested term for the arithmetic sequence.
Find a20 if a1 = 5 and d = 2.
Exercise 18
Find the requested term for the arithmetic sequence.
Find a10 if a1 = -3 and d = -5.
Exercise 19
Find the requested term for the arithmetic sequence.
Find a100 if a1 = 7 and d = 5.
Exercise 20
Find a1 and d for the arithmetic sequence with the given terms.
a100 = 204, a101 = 206
Exercise 21
Find a1 and d for the arithmetic sequence with the given terms.
a11 = 22, a12 = 33
Exercise 22
Find a1 and d for the arithmetic sequence with the given terms.
a20 = -2, a21 = -7.
Exercise 23
How many steps does it take to get from a10 to a15?
Exercise 24
How many steps does it take to get from a100 to a200?
Exercise 25
How many steps does it take to get from am to an (assume m < n)?
Exercise 26
Find a1 and d for the arithmetic sequence with the given terms.
a20 = 13, and a25 = 33