SOLUTION: what is the next number in the sequence 6,3,12,6,14,8,22,?

6,3,12,6,14,8,22,

List the numbers vertically:

 6
 3
12
 6
14
 8
22

Subtract every number from the number just below it and write the
result beside it.  The -3 comes from 3-6 = -3.  The 14 coes from
22-8 = 14   

  6  -3 
  3   9 
 12  -6  
  6   8 
 14  -6  
  8  14
 22 

Do the same for the second column.  Each column will have one
less than the preceding one.  Continue until all the numbers in
the column are the same.  In this case we had to continue until
there was just one number in the 7th column:

  6  -3  12 -27  56 -113  232
  3   9 -15  29 -57  119  
 12  -6  14 -28  62  
  6   8 -14  34 
 14  -6  20 
  8  14 
 22 

Now write another 232 below the 232 in the 7th column.
Then work backwards by adding. For instance the 351 was
gotten by adding the 232 and the 119 just to the left of it,
and writing it under the 119. 


  6  -3  12 -27  56 -113  232
  3   9 -15  29 -57  119  232
 12  -6  14 -28  62  351
  6   8 -14  34 413
 14  -6  20 447
  8  14 467
 22 481 
503

And we see that the next term is 503.

That's the next number you were asking for.
You don't need it, but to get the next term after that,
put another 232 in the 7th column, and work backwards by
adding the same way:

   6   -3   12  -27  56 -113  232
   3    9  -15   29 -57  119  232
  12   -6   14  -28  62  351  232
   6    8  -14   34 413  583 
  14   -6   20  447 996
   8   14  467 1443
  22  481 1910
 503 2391
2894

The general expression can be gotten by substituting 1,2,3,... for n

and the corresponding terms of the sequence for 

and solving the resulting system of equations:



The resulting general equation is:



Edwin