How do you tell whether the sequence 2,4,8,16,.... is arithmetic, geometric or neither?

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Answer 1 · 2016-05-01T15:26:45

Geometric as far as it goes.

It is geometric as far as it goes.

Notice that the ratio between each successive pair of terms is constant:

#4/2 = 2#

#8/4 = 2#

#16/8 = 2#

That these ratios are all the same is sufficient for the given terms to form a geometric sequence with general term:

#a_n = 2^n#

The sequence then continues:

#2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048,...#

The question is in the "...". Any finite number of terms does not determine an infinite sequence.

For example we can match the sequence #2, 4, 8, 16# with a cubic polynomial:

#a_n = 1/3(n^3-3n^2+8n)#

Then we would find that the next terms would be #30, 52, 84,...#