Projects
days until the end of spring semester.days until Snakes on a Plane.
Boxes unpacked
Math article project
Finished mathematical core of article. Next: Write analytical core of article.
Dummit and Foote, Abstract Algebra
Finished section 1.6 (86 to go)
Silverman and Tate, Rational Points on Elliptic Curves
FInished 2.5 (31 to go)
Conway, Functions of One Complex Variable I
Finished section 7.5 (27 to go)
Munkres, Topology
Finished section 21 (60 to go)
Royden, Real Analysis
Finished section 2.4 (97 to go)
Nonfiction book project
Todo list uptodate
Fiction book project
1443 out of 100,000 projected words written.
Top 100 novels of all time
Reading Ulysses
IMDB top 250 films
Tengoku to jigoku next in queue.
Blogroll
This academic life
Academic CoachConfessions of a Community College Dean
Learning Curves
The Little Professor
My Hiding Place
New Kid on the Hallway
One Bright Star
Planned Obsolescence
Tall, Dark, and Mysterious
Math blogs
Ars MathematicaMathForge
MathPuzzle
Think Again
Archives

July 2003
August 2003
September 2003
October 2003
November 2003
December 2003
January 2004
February 2004
March 2004
April 2004
May 2004
June 2004
July 2004
August 2004
September 2004
October 2004
December 2004
January 2005
March 2005
April 2005
May 2005
June 2005
July 2005
August 2005
September 2005
October 2005
November 2005
December 2005
January 2006
February 2006
March 2006
April 2006
May 2006
Vito Prosciutto: Teaching community college math on the road to a PhD.
Wednesday, July 06, 2005
11:10Some math content
Consider the case where we're investing $100/month at 6% annually (which is 0.5% monthly, thus my choice of a 6% rate). So at the end of the first month there's the $100 investment, plus its interest (I=prt) of $.50. At the end of the second month, we've added $100 and we're going to pay interest on $200.50. And the process repeats. That's not terribly helpful. Let's look at the abstract case.
Let the a be the monthly investment amount and x be the monthly interest rate. Now we have the following amounts at the end of each month:
Month 1: ax
Month 2: (ax+a)x=ax^{2}+ax
Month 3: (ax^{2}+ax+a)x=ax^{3}+ax^{2}+ax
Now we see a pattern. We can factor out the a and see that we're left with
[sum from i=1 to n] x^{i}
We just need a formula to express this more succinctly. Trying some concrete numbers to see what happens helps when I pick x=10. Then we're talking about (letting the lower limit be 0 rather than 1) a number which is all 1's. It occurs to me that this can be described by taking x^{n+1}1 and dividing by x1. I double check this with some polynomial long division to make sure that it is in fact the expected x^{n}+...+x+1 and now I know that I can find the total saved after n months by taking
P=a[((1+(r/12))^{n+1}1)/(r/12)1]
where r is the annual savings rate. We add 1 because x will be 1 plus the monthly savings rate. The minus 1 at the end is to take care of the fact that our expression is actually for the sum from 0 to n but we don't want to include that 0 term (which reflects another contribution to the pot at the end of the savings term which hasn't had interest applied to it).
Next I'll have to tackle the installment loan payment amount formula. How hard can it be?