Expert: Josh - 2/11/2007

Question
I would like to know the name of this series and how I could get more information on it. Here is an example:

3, 21, 75, 237

In this series the difference between one term and the next greater term forms a geometric sequence so in order to find a certain term one needs to use the formula used to calculate the sum of a geometric sequence. For example, 21-3 is 18, 75-21 is 54, 237-75 is 162. All of these are in multiples of 3 which form a geometric series and in order to find a term the sum of this geometric series must be calculated. I know what that formula is and how to calculate but I need to know if such a sequence is covered in math and the name of it and where I can get more information. Thank You.

Answer
Hello Pedram,

I don't know if a name is ascribed to this sequence. However, we can describe this sequence as being a geometric sequence constructed upon the first order difference.

Given t[n], we first form the difference sequence:
 d[n-1]=t[n]-t[n-1] for n>=1.
Then,
 d[n]=r^n*d[0] for n>=0, where r=3, d[0]=18.

We have a "first order difference equation":
t[n]=t[n-1]+d[n-1]
   =r^(n-1)*d[0]+t[n-1]
By recursion,
t[n-1]=t[n-2]+d[n-2]
   =r^(n-2)*d[0]+t[n-2]
...
t[2]=t[1]+d[1]
   =r*d[0]+t[1]
t[1]=t[0]+d[0]
   =d[0]+t[0]
Thus,
t[n]=K+d[0]*(1+r+r^2+...+r(n-1)), where constant K=t[0].

We can rewrite this as t[n]=K+d[0]*(1-r^n)/(1-r).
This is an exponential model -- the nth term is precisely some multiple of the geometric sum associated with the difference sequence, plus an offset K.

This might be used in financial mathematics to model time series.