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A sequence is arithmetic if the subtraction between any pair of consecutive terms is constant.


A sequence is arithmetic if the ratio between any pair of consecutive terms is constant.


Let's pick consecutive couples and check their difference and ratio:


First couple: first and second term

Difference:
0 - (-2) = 0+2 = 2

Ratio:
(0)/(-2) = 0


Second couple second and third terms:

Difference:
2 - 0 = 2

Ratio:
(2)/(0) = \text{undefined}


So, we can see that the difference remained constant (is always 2), while the ratio isn't even defined. So, this is an arithmetic sequence.


As a corollary, you can see that a geometric sequence can never contain 0. Otherwise, when checking consective ratios, that 0 will eventually be a denominator and break the procedure.

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