Question

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

Answer 1

This sequence is neither.

Explanation:

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

`8-3 = 5 != 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:

`8/3 = 64/24 != 27/24 = 9/8`

So this sequence has no common ratio.

It is therefore neither an arithmetic nor geometric sequence.

`color(white)()`
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:

`color(blue)(3), 8, 9, 12`

Then write down the sequence of differences between successive terms:

`color(blue)(5), 1, 3`

Then write down the sequence of differences of those differences:

`color(blue)(-4), 2`

Then write down the sequence of differences of those differences:

`color(blue)(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+5n-5-2n^2+6n-4+n^3-6n^2+11n-6`

`=n^3-8n^2+22n-12`