# 52 Week Reverse Savings Plan

Yesterday I came across this picture taken by one of my former high school students:

I didn’t really know what the 52 week reverse savings plan was, but based on her hashtags and the amount of money she deposited yesterday it seemed reasonable that the goal was to save money each week (there are, after all 52 weeks in a year) and that she would decrease the amount of money she was depositing into her savings account by one dollar each week.  (Turns out I was right).

I can’t be certain, but I think the idea behind this type of savings plan is that you capitalize on the idea that at the beginning of the year, right after you’ve made your New Year’s resolution, you’re more likely to set aside larger amounts of money for the program and as the program continues, you can talk yourself into saving the next week, because its less then you set aside the week before.  I’d venture a guess that the reverse of this 52 week reverse savings plan would not be as effective.

Her photo made me think of a story I used to tell my Pre-Calculus and Algebra II students when we began talking about series of numbers.  The story is this (and I think its loosely based on a true story.  Read it here):

Carl Friedrich Gauss is a famous mathematician, and as is true with other young geniuses, his elementary school teachers found young Carl to be quite annoying and unruly.  Why?  You might ask.  Well, for two reasons really.  First, Carl could finish the work intended to take 30 minutes in 5, thus spending the remaining 25 minutes doing what young children do when they’re bored.  Second, Carl seemed to be able to outsmart his teachers in almost everything.  One day at school the same scenario that had been playing out for days once again played out in young Carl’s classroom–his teacher had given an assignment and Carl had finished in a fraction of the time the assignment was meant to take.  As he began to distract and disrupt his other classmates, his teacher had a brilliant idea!  She called Carl up to her desk and told him to add all of the integers from 1 to 100.

Imagine his teacher’s surprise (and probably frustration!) when Carl came back to her desk a mere minute later with the correct answer!

When questioned about what he had done he laid out the following pattern for the teacher:

So,

But now I’ve added the numbers from 1-100 twice, so to account for this I really need to write:

Isn’t that clever?

And, can you tell how this relates to the Instagram pic posted by one of my former students?  It seems to me that it would be reasonable to ask how much money she will have saved by the end of 2014.  One way we could answer this question would be to add money deposited each week:

52+51+50…+3+2+1

But, that seems a little tedious and thanks to Carl Gauss we can do this more efficiently.  Namely:

(Similar to my M&M posts (here, here, and here), I smell a series (ha!-get it, series?) of posts related to this topic. For example, how many weeks does it take to save half of the money from the 52 week challenge?  If my student is depositing this money into a savings account, then she’s earning interest.  If she leaves the money in the account until she goes to college in two years, how much money will she have?  Is the amount really all that different if she only saves for half the year?  Or every other week? . . . the possibilities are limitless (ha! ha!-get it, limitless?)) (Check out the second post in the series here.)