Answer:
- Neither
- Arithmetic
- Geometric
Explanation:
You want to know the nature of the given sequences:
- 4, 1, 1, 4, ...
- 7, 13, 19, 25, ...
- 250, 50, 10, 2, ...
Differences
Looking at differences between terms can tell you a lot about the nature of a sequence.
When first differences are constant, the sequence is arithmetic.
When first differences are not constant, but have a common ratio, the sequence is geometric. (The sequence terms will have the same common ratio.)
When second differences are constant, the sequence is quadratic. (Not applicable here.)
4, 1, 1, 4
First differences are -3, 0, 3. These are not constant. However, second differences are 0 -(-3) = 3 and 3 -0 = 3. These are constant, so this sequence is quadratic, neither arithmetic nor geometric.
7, 13, 19, 25
First differences are 6, 6, 6. They are constant, so this sequence is arithmetic.
250, 50, 10, 2
First differences are -200, -40, -8. These are not constant, but have a common ratio of 1/5 — just as the terms of the sequence do.
This sequence is geometric.
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Additional comment
The first differences are computed by subtracting the term before from each term. 1 -4 = -3; 1 -1 = 0, 4 -1 = 3 for the first sequence. The ratio is found by dividing a term by the term before: 50/250 = 1/5, for example.
For common difference d and first term a1, the general term of an arithmetic sequence is ...
an = a1 +d(n -1)
For common ratio r, the general term of a geometric sequence is ...
an = a1·r^(n-1)
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