How do you tell whether the sequence 2,4,8,16,.... is arithmetic, geometric or neither?
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
#a_n = 1/3(n^3-3n^2+8n)#
Then we would find that the next terms would be