Algebra Patterns - Add add subtract

Let's re-state the challenge.

What is the value of the following expression where all the whole numbers 1 through 99 are included and the repeated operations are add, add, subtract?

0 + 1 + 2 3 + 4 + 5 6 + 7 + 8 9 + . . . + 97 + 98 99

 

I chose this investigation because I thought it would be a nice introduction to doing write-ups. 

When I began this investigation I wanted to write a formula to recognize the pattern.  I tried using consecutive numbers, but found that did not solve the problem.  I went to a spreadsheet and originally tried to enter a new formula on each line hoping that the spreadsheet might recognize the repeating pattern or some magic formula would appear.  This was unsuccessful.

Then I tried doing the math by pencil paper method.  I still did not see a pattern. 

Only when I broke up the segments (add add subtract), did I find a pattern.  The individual segments increased by three.  I was then able to use a formula in a spreadsheet to complete my calculations and arrive at the end value of 1584.

Click here to view spreadsheet.

Included in the spreadsheet is a graph that represents the expression y=3x.  By plotting only the early values of x (0,3,6,9) and then plotting ending value of x (96) we can see that the graph appears to be a straight line.  As we continue and fill in these values we see that indeed it is a straight line.