Example 4.  Application to number theory.
4 (a).  Find the formula for the sum of the first  n  integers.
4 (b).  Find the formula for the sum of the squares of the first  n integers.  

Solution 4.

4 (a).  Find the formula for the sum of the first  n  integers.

Notice that the divided difference table has zeros for the 2nd and higher order differences.  

1
3	2
6	3
10	4		0
15	5		0	0

Thus the formula for the sum of the first n integers is:        

Compare this formula with Mathematica's knowledge.    

4 (b).  Find the formula for the sum of the squares of the first  n  integers is:

Notice that the divided difference table has zeros for the 3rd and higher order differences.

0
1	1
5	4
14	9
30	16			0
55	25			0	0

Thus the formula for the sum of the squares of the first  n  integers is:        

Compare this formula with Mathematica's knowledge.    

(c) John H. Mathews 2004