Determine whether the sequence is arithmetic or not.
1, 2, 3, 4, 5,...
Answer
The step up from one term to the next term is always 1.
Equivalently, the step down from one term to the previous term is always 1:
This is an arithmetic sequence.
Example 2
Determine whether the sequence is arithmetic or not.
10, 13, 16, 20,...
Answer
The step up from one term to the next isn't always the same.
That means this is NOT an arithmetic sequence.
Example 3
Determine whether the sequence is arithmetic or not.
1, 3, 5, 7, 9,...
Answer
Look at the difference between successive terms:
3 – 1 = 2
5 – 3 = 2
7 – 5 = 2
9 – 7 = 2
Since we always get 2 as the difference, this is an arithmetic sequence.
Example 4
Determine whether the sequence is arithmetic or not.
15, 21, 26, 30,...
Answer
Look at the difference between successive terms:
21 – 15 = 6
26 – 21 = 5
30 – 26 = 4
Since we don't get the same difference every time, this is not an arithmetic sequence.
Example 5
Determine whether the sequence is arithmetic or not.
0, 10, 20, 30,...
Answer
Look at the step up from one term to the next.
Since the step is always 10, this IS an arithmetic sequence.
Example 6
Determine whether the sequence is arithmetic or not.
1, -1, 1, -1,...
Answer
The step from one term to the next alternates between -2 and + 2.
Since the step isn't always the same, this is NOT an arithmetic sequence.
Example 7
Determine whether the sequence is arithmetic or not.
5, -10, 15, -20,...
Answer
Look at the difference between successive terms:
-10 – 5 = -15
15 – (-10) = 25
-20 – 15 = -35
Since the difference between successive terms isn't always the same, this is NOT an arithmetic sequence.
Example 8
Determine whether the sequence is arithmetic or not.
0, -2, -4, -6,...
Answer
To get from one term to the next, we subtract 2 (or add -2, if you prefer).
Since the step size is always the same, this IS an arithmetic sequence.
Example 9
Write the first four terms of the arithmetic sequence with
a1 = 10 and d = 7
Answer
We start at 10 and step up 7 to get to the next term.
The first four terms are 10, 17, 24, 31.
Example 10
Write the first four terms of the arithmetic sequence with
a1 = 5 and d = -2
Answer
We start at 5 and step down 2 each time.
The first four terms are 5, 3, 1, -1.
Example 11
Write the first four terms of the arithmetic sequence with
a1 = -1 and d = -3
Answer
We start at -1 and step down 3 each time.
The first four terms are -1, -4, -7, -10.
Example 12
How many steps does it take to get from
a1 to a6?
Answer
It takes 5 steps.
Example 13
How many steps does it take to get from
a1 to a75?
Answer
It takes 74 steps. From a1 to a2 is 1 step. From a1 to a3 is 2 steps. From a1 to a4 is 3 steps.
Continuing this pattern, from a1 to a75 must be 74 steps.
Example 14
How many steps does it take to get from
a1 to an?
Answer
Continuing the pattern, from a1 to an takes (n – 1) steps.
Example 15
Find the requested term for the arithmetic sequence.
Find a8 if a1 = 20 and d = 3.
Answer
It takes 7 steps to get from a1 to a8. The step size is d = 3, so
a8 = a1 + 7d
= 20 + 7(3)
= 41.
Example 16
Find the requested term for the arithmetic sequence.
Find a11 if a1 = 1 and d = -1.
Answer
It takes 10 steps of size d = -1 to get from a1 to a11, so
a11 = a1 + 10d
= 1 + 10(-1)
= -9.
Example 17
Find the requested term for the arithmetic sequence.
Find a20 if a1 = 5 and d = 2.
Answer
It takes 19 steps of size 2 to get from a1 to a20, so
a20 = a1 + 19d
= 5 + 19(2)
= 43.
Example 18
Find the requested term for the arithmetic sequence.
Find a10 if a1 = -3 and d = -5.
Answer
It takes 9 steps to get from a1 to a10. If each step has size -5, then
a10 = a1 + 9d
= -3 + 9(-5)
= -48.
Example 19
Find the requested term for the arithmetic sequence.
Find a100 if a1 = 7 and d = 5.
Answer
It takes 99 steps to get from a1 to a100. If the step size is d = 5, then
a100 = a1 + 99d
= 7 + 99(5)
= 502.
Example 20
Find a1 and d for the arithmetic sequence with the given terms.
a100 = 204, a101 = 206
Answer
The step size is
d = a101 – a100 = 2.
To get from a1 to a100 is 99 steps of size 2, so
a1 + 99d = a100
a1 + 99(2) = 204
a1 = 6.
We have a1 = 6 and d = 2.
Example 21
Find a1 and d for the arithmetic sequence with the given terms.
a11 = 22, a12 = 33
Answer
The step size is
d = a12 – a11 = 11.
There are 10 steps from a1 to a11, so
a1 + 10d = a11
a1 + 10(11) = 22
a1 = -88.
We have a1 = -88 and d = 11.
Example 22
Find a1 and d for the arithmetic sequence with the given terms.
a20 = -2, a21 = -7.
Answer
The step size is
d = a21 – a20 = -5.
It takes 19 steps to get from a1 to a20, so
a1 + 19d = a20
a1 + 19(-5) = -2
a1 = 93
We have a1 = 93 and d = -2.
Example 23
How many steps does it take to get from a10 to a15?
Answer
It takes 5 steps.
Example 24
How many steps does it take to get from a100 to a200?
Answer
It takes 1 step to get from a100 to a101, 2 steps to get from a100 to a102, and so on. It must take
100 steps to get from a100 to a200.
Example 25
How many steps does it take to get from am to an (assume m < n)?
Answer
Suppose you have a staircase with n stairs, labeled from 1 to n. If you're standing on the mth stair,
there are (n – m) stairs left before you get to the top.
This translates nicely to sequences. If you're on the mth term of a sequence, and you want to get to the nth term, that means you have to take
n – m
steps.
Example 26
Find a1 and d for the arithmetic sequence with the given terms.
a20 = 13, and a25 = 33
Answer
We have
a25 – a20 = 20.
There are 5 steps from a20 to a25, so we divide by 5 to get
d = 4.
Then we proceed as before:
a1 + 19d = a20
a1 + 19(4) = 13
a1 = -63.
Once we know we have an arithmetic sequence, if we start at a1 and take (n – 1) steps of size d, we end up on term an.
We can write this idea in symbols as
a1 + (n – 1)d = an.
This formula has three unknowns: a1, n and d. Like a buy two, get one free deal on cups of key lime pie yogurt, any two pieces of information about an arithmetic sequence tells us the third and final piece of information. So, if we know a1 and d we can find any term an we like. We can also work backwards from d and some term an (which gives us n) to find a1.