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" (1³+2³+3³+...)

	n=1	n=2	n=3	n=4	n=5	Conjectured formula:
n     2i i=1
n     i i=1
n     i3 i=1

Extra Challenge: Can your group prove the formula for

n     2i i=1

(and thus for

n     i i=1

) ?

Suggestion: write the first sum as the sum of two rows of numbers:

	1	+	2	+	3	+	...	+	n-1	+	n
+	n	+	n-1	+	n-2	+	...	+	2	+	1
=

What do you notice as you sum these numbers up?

(The great mathematician Karl Friedrich Gauss (1777-1855) discovered this "trick" in grade school!)