Sigma Detectives  Your name:______________ 
Names of fellow students in your group: 
Fill in the entires of the following chart. Then, make a conjecture
(educated guess) about what formulas might give the sum of the
first n even integers, the sum of the first n integers,
and the sum of the first n "cubes"
(1^{3}+2^{3}+3^{3}+...)
n=1  n=2  n=3  n=4  n=5  Conjectured formula:  

 
 

 (and thus for 
 ) ? 
1  +  2  +  3  +  ...  +  n1  +  n  
+  n  +  n1  +  n2  +  ...  +  2  +  1 
= 
What do you notice as you sum these numbers up?
(The great mathematician Karl Friedrich Gauss (17771855) discovered this "trick" in grade school!)