Expert: Scott A Wilson - 9/7/2010

Question
2,11:   4,67:  5,128:  3, ?  Please can you tell me what is the relationship between each set of numbers and what the missing number is?

Answer
I created a difference table in Excel.
That is,
2   11         
3   0   -11      
4   67   67   78   
5   128   61   -6   -84

Note that we are looking for a value to put where there is a 0 so that the -84 out on the right end becomes 0.

To get the 3rd column, it is the difference of elements in the 2nd colmnu.  To get the 4th column, it is the difference of elements in the 3rd column.  To get the 5th column, it is the difference of elements in the 3rd column.

Thus, the -11 = 0 - 11; 67 = 67- 0; 61 = 128 - 67; 78 = 67 - -11;
-6 = 61 - 67; and -84 = -6 - 78.

Guessing it to be around 30, we get
2   11         
3   30   19      
4   67   37   18   
5   128   61   24   6

Since 6 is closer to 0 than -84, 30 is closer to the right answer than 0 was.  Since it has a much smaller magnitude, 30 is much closer to the answer than 0 was.  Since 6 is positive and -84 is negative, the answer is in between.

I'll try 29, giving
2   11         
3   29   18      
4   67   38   20   
5   128   61   23   3

Next, why not try 28.  I think that's the answer.


Now as far as the relationship, here is how it was found:
Take the original values.  That is,
x   y   A   B   C
2   11
3   28   17
4   67   39   11
5   128   61   11   0
where A is the difference in y's over the difference in the x's,
B is the difference in the A's over the difference in x and the x two before, and C is the difference in the B's over the difference in the x and the x three before.  That is, 11 = (30-17)/(3-1) and
0 = (11-11)/(5-2).
              
x   This column has the column above -11x^2.
x   y   A   B   C
2   -33
3   -71   -38
4   -109   -38   0
5   -147   -38   0   0
  
This column has the one above + 38 times the number in the first column.
x   y   A   B   C         
2   43
3   43   0
4   43   0   0
5   43   0   0   0

So the equation can be said to be 11x^2 - 38x + 43.
This should be taken and checked to make sure it is correct.