Determine whether the sequence is arithmetic, geometric, both, or neither.
If the sequence is arithmetic, find d. If the sequence is geometric, find r.
5, 5, 5, 5,...
Answer
To determine if the sequence is arithmetic, look at the differences between successive terms. Since all the terms are 5, all the differences are
5 – 5 = 0.
The sequence is arithmetic with d = 0.
To determine if the sequence is geometric, look at the ratios between successive terms. Since all the terms are 5, all the ratios are
The sequence is geometric with r = 1.
This sequence is both arithmetic and geometric.
Example 2
Determine whether the sequence is arithmetic, geometric, both, or neither.
If the sequence is arithmetic, find d. If the sequence is geometric, find r.
1, 2, 5, 10, 17,...
Answer
To determine if the sequence is arithmetic, look at the differences between successive terms.
2 – 1 = 1
5 – 2 = 3
10 – 5 = 5
17 – 10 = 7
The differences are odd numbers, which is interesting, but they aren't all the same. The sequence isn't arithmetic.
To determine if the sequence is geometric, look at the ratios between successive terms.
The ratios aren't all the same, so the sequence isn't geometric either.
Example 3
Determine whether the sequence is arithmetic, geometric, both, or neither.
If the sequence is arithmetic, find d. If the sequence is geometric, find r.
81, -27, 9, -3,...
Answer
To determine if the sequence is arithmetic, look at the differences between successive terms.
(-27) – 81 = -108
9 – (-27) = 36
The differences aren't all the same, so the sequence isn't arithmetic.
To determine if the sequence is geometric, look at the ratios between successive terms.
The ratios are all the same. This sequence is geometric with .
Example 4
Determine whether the sequence is arithmetic, geometric, both, or neither.
If the sequence is arithmetic, find d. If the sequence is geometric, find r.
100, 90, 80, 70, ...
Answer
To determine if the sequence is arithmetic, look at the differences between successive terms.
90 – 100 = -10
80 – 90 = -10
70 – 80 = -10
The differences are all the same. The sequence is arithmetic with
d = -10.
To determine if the sequence is geometric, look at the ratios between successive terms.
The ratios aren't all the same, so the sequence isn't geometric.
Example 5
Determine whether the sequence is arithmetic, geometric, both, or neither.
If the sequence is arithmetic, find d. If the sequence is geometric, find r.
100, 50, -25, 12.5, 6.25, -3.125, ...
Answer
To determine if the sequence is arithmetic, look at the differences between successive terms.
50 – 100 = -50
-25 – 50 = -75
The differences aren't all the same, so the sequence isn't arithmetic.
To determine if the sequence is geometric, look at the ratios between successive terms.
The ratios aren't all the same, so the sequence isn't geometric.
Example 6
Is it possible for a sequence to be both arithmetic and geometric?
Answer
Yes, because we found an example above:
5, 5, 5, 5,....
Similarly, any sequence
c, c, c, c,...
where c is a constant will be arithmetic with d = 0 and geometric with r = 1. It turns out that this is the only type of sequence which can be both arithmetic and geometric.
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