# How do you tell whether the sequence 3,8,9,12,.... is arithmetic, geometric or neither?

May 30, 2016

This sequence is neither.

#### Explanation:

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

$8 - 3 = 5 \ne 1 = 9 - 8$

So this sequence has no common difference.

A geometric sequence has a common ratio between successive terms, but in our example we find:

$\frac{8}{3} = \frac{64}{24} \ne \frac{27}{24} = \frac{9}{8}$

So this sequence has no common ratio.

It is therefore neither an arithmetic nor geometric sequence.

$\textcolor{w h i t e}{}$
Bonus

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:

$\textcolor{b l u e}{3} , 8 , 9 , 12$

Then write down the sequence of differences between successive terms:

$\textcolor{b l u e}{5} , 1 , 3$

Then write down the sequence of differences of those differences:

$\textcolor{b l u e}{- 4} , 2$

Then write down the sequence of differences of those differences:

$\textcolor{b l u e}{6}$

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 = color(blue)(3)/(0!) + color(blue)(5)/(1!)(n-1) + color(blue)(-4)/(2!)(n-1)(n-2) + color(blue)(6)/(3!)(n-1)(n-2)(n-3)

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

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