Question

You are having a discussion about sequences with your classmate. She insists the the sequence 2,3,5,8,12 must be either arithmatic or geometric. Is she correct or incorrect? Explain.

Answer 1

The sequence is, in fact, quadratic. It is described by the equation
.. a[n] = (n*(n -1))/2 +2

First differences are increasing, so the sequence will not be arithmetic. An arithmetic sequence has constant first differences.

Second differences are constant, so the sequence will not be geometric. A geometric sequence will have first-, second-, third-differences, and those to any level, that have the same constant ratio as the terms of the original sequence.