Whether the sequence 3,8,9,12,.... is arithmetic, geometric or neither

Question

Kendall Ponce · Accepted Answer

An arithmetic sequence has a common difference between successive terms, but in our example we find:

8 - 3 = 5 \neq 1 = 9 - 8

So this sequence has no common difference.
A geometric sequence has a common ratio between successive terms, but in our example we find:

\frac{8}{3} = \frac{64}{24} \neq \frac{27}{24} = \frac{9}{8}

So this sequence has no common ratio.
It is therefore neither an arithmetic nor geometric sequence.

puntdald8 · Accepted Answer

Given any finite sequence, it is possible to construct a polynomial expression for a general term that matches the given sequence.
In our example, write down the sequence:
3,8,9,12
Then write down the sequence of differences between successive terms:
5,1,3
Then write down the sequence of differences of those differences:
−4,2
Then write down the sequence of differences of those differences:
6
Having arrived at a constant sequence we can write down a formula for a general term using the first term of each of these sequences as coefficients:

a_{n} = \frac{3}{0!} + \frac{5}{1!} (n - 1) + \frac{- 4}{2!} (n - 1) (n - 2) + \frac{6}{3!} (n - 1) (n - 2) (n - 3)

= 3 + 5 n - 5 - 2 n^{2} + 6 n - 4 + n^{3} - 6 n^{2} + 11 n - 6

= n^{3} - 8 n^{2} + 22 n - 12

Whether the sequence 3,8,9,12,.... is arithmetic, geometric or neither

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