What kind of sequence is 8, 12, 18, 27,...?
1 Answer
As far as given, it is a geometric sequence with common ratio
Explanation:
Given:
8, 12, 18, 27
Note that:
12/8 = 18/12 = 27/18 = 3/2
So the given terms have a common ratio
The general term of a geometric sequence can be written:
a_n = a * r^(n-1)
where
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
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)