Find the Formula: 1, 6, 19, 44, 85

Date: 01/31/2002 at 12:16:08
From: Krisna
Subject: Sequences

I am currently studying sequences, and am having particular problems 
with one sequence. This sequence is as follows:

   1, 6, 19, 44, 85

I think you will find the difference to be 4. 

Please include the different steps needed to find the formula.

Thank you so much.

Date: 02/03/2002 at 06:24:04
From: Doctor Floor
Subject: Re: Sequences

Hi, Krisna,

First make consecutive sequences of differences:

  1   6  19  44  85
    5  13  25  41
     8   12  16
       4   4

The third differences are constant, which suggests a third degree 
formula, hence:

  f(x) = ax^3 + bx^2 + cx + d

We can use this to extend the sequence two places to the left, to find 
f(0) and f(-1), which are attractive for yielding simple equations in 
the system of equations later on:

-1  0  1   6  19  44  85
   1  1  5  13  25  41
     0  4  8  12  16
       4  4  4   4


Now I substitute n = -1, 0, 1, 2, yielding the equations

  x = -1:  -a + b - c + d = -1
  x = 0:   d=0
  x = 1:   a + b + c + d = 1
  x = 2:   8a + 4b + 2c + d = 6

Adding the equations yielded by x = -1 and x = 1 gives

  2b + 2d = 0

Together with d = 0 from x = 0, this shows that b = 0. So we have 
d = 0 and b = 0. Substituting this into the equations for x = 1 and 
x = 2, we find

  a + c = 1
 8a + 2c = 6

It is now not difficult to find the solution a = 2/3 and c = 1/3. I 
leave the details to you.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/