Expert: Scott A Wilson - 9/16/2009

Question
Learning conjectures; Question I don't get.
Write a rule in terms of "n" for the sum of the first "n" even positive integers.  I see that the sum would be n + all the even integers that come before it.  I can not tell if it is possible to write an equation using n to express this.

Answer
The list of positive, even integers is 2, 4, 6, 8, 10, 12, ...
The sum is 2,
2+4=6
2+4+6=12
2+4+6+8=20
2+4+6+8+10=30

Use these values to form a difference table.
i sum
1 2
2 6 (6-2)/(2-1) = 4
3 12 (12-6)/(3-2) = 6 (6-4)/(3-1) = 1
4 20 (20-12)/(4-3) = 8 (8-6)/(4-2) =  1
5 30 (30-20)/(5-4) = 10 (10-8)/(5-3) = 1

Since it only goes out to levels {the third column would have (1-1)/(4-1) and (1-1)/(5-2) }
it can be fit by taking n to the power 2.

That is, subtract of x^2 from the sum since there is a 1 in 2nd column.
That gives
1 2-1=1
2 6-3=2
3 12-9=3
4 20-16=4
5 30-25=5

Upon inspection of this data, it looks like 1, 2, 3, 4, 5 is linear,
so there is also an x in the equation.

Looking at x² + x = x(x+1), we get
1 2
2 6
3 12
4 20
5 30

Now that looks like the starting set of data, so there is our equation.
That is, y = x(x+1) for x=1,2,3,4,5...


If the x column, instead of starting 1,2,3,4,5... started 2,4,6,8,10,...
the equation would have an x/2 put in for x.
That is, the new equation would be (x/2)² + x/2 = (x²+2x)/4 = x(x+2)/4
for x=2,4,6,8,10...