Question

What kind of sequence is `8, 12, 18, 27,...`?

Answer 1

As far as given, it is a geometric sequence with common ratio `3/2`

Explanation:

Given:

`8, 12, 18, 27`

Note that:

`12/8 = 18/12 = 27/18 = 3/2`

So the given terms have a common ratio `3/2` and constitute a geometric sequence.

The general term of a geometric sequence can be written:

`a_n = a * r^(n-1)`

where `a` is the initial term and `r` the common ratio.

So in our example:

`a_n = 8(3/2)^(n-1)`

Alternatively, we can write a recursive rule of the form:

`{ (a_1 = a), (a_(n+1) = r a_n " for " n >= 1) :}`

In our example:

`{ (a_1 = 8), (a_(n+1) = 3/2 a_n " for " n >= 1) :}`

Footnote

Actually, since we have not been told what kind of sequence this is and have only been given the first `4` terms, the following terms could be anything. For example, we could match it with a cubic formula using the method of differences:

Write down the initial sequence:

`color(blue)(8), 12, 18, 27`

Write down the sequence of differences between consecutive pairs of terms:

`color(blue)(4), 6, 9`

Write down the sequence of differences of those differences:

`color(blue)(2), 3`

Write down the sequence of differences of those differences:

`color(blue)(1)`

Having reached a constant sequence (albeit of just one term), we can use the initial term of each of these sequences as coefficients to give us a formula:

`a_n = color(blue)(8)/(0!)+color(blue)(4)/(1!)(n-1)+color(blue)(2)/(2!)(n-1)(n-2)+color(blue)(1)/(3!)(n-1)(n-2)(n-3)`

`color(white)(a_n) = 1/6(n^3 + 17n + 30)`