I assume you want it in the form 

First we find the general term by assuming its general term
can be expressed as a polynomial.  To find the smallest
possible degree it can have we write the four given numbers in
a line like this

1   6   13   22 

Then we find a line of differences by subtracting each number 
from the one just to the right of it and writing the difference 
between and below the numbers subtracted, like this:

1   6   13   22
  5   7    9

5, 7, and 9 are not all the same number, so we do the same to 
the bottom line:

1   6   13   22
  5   7    9
    2    2

Those are the same, since they are both 2's.  Since it requires 2 lines
of differences to get them all to be the same number, we assume the general
term is a polynomial of degree 2:



We substitute n=1,2, and 3

  





We solve that system and get

, , and 

So,





So we write:



Note: The series could also be written in the form:



where the index (or "dummy variable") n starts at 0.  Then the
system of equations would be easier to solve.  But I am guessing 
that your teacher probably wanted the summation index to start at
1, not 0.

Edwin