State whether the following sequence is arithmetic,geometric,or neither. 0,1,4,9,16,25,36,...
See the Related Questions below.
Arithmetic
To determine whether the given sequence 0, 1, 4, 9, 16, 25, 36,... is arithmetic, geometric, or neither, we need to examine the differences between consecutive terms.
Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant.
Geometric Sequence: In a geometric sequence, the ratio between consecutive terms is constant.
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 observe that the differences are not constant. They are increasing by 2 each time. Therefore, the given sequence is not an arithmetic sequence.
Now let's examine the ratios between consecutive terms:
1 / 0 = undefined
4 / 1 = 4
9 / 4 = 2.25
16 / 9 = 1.77
25 / 16 = 1.56
36 / 25 = 1.44
Since the ratios between consecutive terms are not constant, the given sequence is also not a geometric sequence.
Hence, the given sequence 0, 1, 4, 9, 16, 25, 36,... is neither an arithmetic nor a geometric sequence.