State whether the following sequence is arithmetic, geometric, or neither.
0, 1, 4, 9, 16, 25, 36, .... (Points : 1)
Well, this sequence seems to be quite rebellious and doesn't fit into any of the traditional categories. It's not an arithmetic sequence because the gaps between the terms are not consistent. It's also not a geometric sequence because the ratios between the terms are not consistent. So, let's just call it a "neither" sequence or perhaps a "mysterious" sequence.
To determine whether a sequence is arithmetic or geometric, we need to check the differences between consecutive terms.
In the given sequence: 0, 1, 4, 9, 16, 25, 36, ....
The differences between consecutive terms are:
1 - 0 = 1
4 - 1 = 3
9 - 4 = 5
16 - 9 = 7
25 - 16 = 9
36 - 25 = 11
Since the differences are not constant, the sequence is neither arithmetic nor geometric.
To determine whether the given sequence is arithmetic, geometric, or neither, we need to look for a pattern.
In an arithmetic sequence, the difference between consecutive terms is always the same. In a geometric sequence, the ratio between consecutive terms is always the same.
Let's find the differences between consecutive terms in the given sequence:
1 - 0 = 1
4 - 1 = 3
9 - 4 = 5
16 - 9 = 7
25 - 16 = 9
36 - 25 = 11
Looking at the differences, we can see that they are not the same. Therefore, the sequence is not arithmetic.
Now let's find the ratios between consecutive terms:
1/0 = undefined
4/1 = 4
9/4 = 2.25
16/9 = 1.777...
25/16 = 1.5625
36/25 = 1.44
Looking at the ratios, we can see that they are not the same. Therefore, the sequence is not geometric either.
Hence, the given sequence is neither arithmetic nor geometric.
Arithmetic sequence:
the difference between consecutive terms is constant.
Example:
1,3,5,7,9
difference:
2,2,2,2
Geometric sequence:
the ratio between consecutive terms is constant.
Example:
1,3,9,27,81,243
ratio:
3,3,3,3,3
For the given example,
0,1,4,9,16,25,36...
Difference:
1,3,5,7,9,11...
ratio
*,4, 9/4, 16/9, 25/16, 36/25 ...
So is the sequence arithmetic, geometric, or neither?