math and tourette's part II
Question:
Chuck Thayer wrote: > There’s actually an intuitive picture that might give an idea of how > somebody cooked up that "short form" formula. My guess is that your > solution may have come from this line of reasoning. It is fairly > simple, and it is amazing that it gives a single formula, namely that: > 1+2+…+N = N(N+1)/2 > Here’s where Randall comes in. As the arbiter of correctness, he will > now be compelled to either produce the inductive proof for this formula > or tell us that it is too trivial to bother with it here. Since we are > in the midst of the Holiday season, let’s hope he does the > latter.<wink> I am sure that Randall has seen not only seen this proof, > but done it as part of his training.
I was within a nanosecond of skipping to the next post when my name caught my eye. You wrote an awefully long post with a lot of steps for a pretty simple math problem. Legend has it that Euler at some very early age was pestering his teacher for mor emath problems to solve. To get him to shut up the teacher assigned him to add the numbers 1 to 100. He returned with an answer in less than a minute. — I | Randall Bart mailto:Barti…@usa.spam.net L |/ Math Genius o | Barti…@worldnet.att.spam.net Barti…@hotmail.spam.com v | 1-818-985-3259 Please reply without spam e | Y |/ Panic in the Year Zero Zero: http://members.aol.com/PanicYr00 o | u |/ Have you solved http://members.aol.com/PanicYr00/Sequence.html
Response:
RandallBart wrote:
: I was within a nanosecond of skipping to the next post when my name : caught my eye. You wrote an awefully long post with a lot of steps for : a pretty simple math problem. Legend has it that Euler at some very : early age was pestering his teacher for mor emath problems to solve. To : get him to shut up the teacher assigned him to add the numbers 1 to : 100. He returned with an answer in less than a minute. actually, it was Gauss. 
it’s a great story. my number theory professor loves to tell it to all of his classes. -Beth "razor sharp tongue in cheek, poking in your open sores" *************************************************************************** *** DoD#4508, AMA#542204, NGG resident fluid dynamics engineer check it out: http://home.cwnet.com/beffie
Response:
Posted and mailed. Hi, Everyone OK. OK. You knew that I couldn’t leave this one alone.<g> There’s actually an intuitive picture that might give an idea of how somebody cooked up that "short form" formula. My guess is that your solution may have come from this line of reasoning. It is fairly simple, and it is amazing that it gives a single formula, namely that: 1+2+…+N = N(N+1)/2 If N is an even number, the solution is fairly easy to envision. 1+2+…+N = 1+N + 2+(N-1) + … + (N/2)+(N/2)+1 = (N+1) + (N+1) + … + (N+1) <—- N/2 terms = N(N+1)/2 Example: N=8. 1+2+…+8 = (1+8) + (2+7) + (3+6) + (4+5) = 9 + 9 + 9 + 9 = 4*9 = (8/2) * 9 = 8*9/2 If N is odd, it is a little more gruesome. Those with delicate sensibilities may not want to watch. 1+2+…+N = 1+N + 2+(N-1) + …+ [(N-1)/2 + (N+1)/2 + 1] + (N+1)/2 = (N+1) + (N+1) + … + (N+1) + (N+1)/2 <– last term is an orphan = (N-1)/2 * (N+1) + (1/2)*(N+1) = [ N/2 - 1/2 + 1/2 ] * (N+1) <– distributive law, remember that? = N(N+1)/2 Example: N=7. 1+2+…+7 = 1+7 + 2+6 + 3+5 + 4 = 8 + 8 + 8 + 8/2 = [3 + 1/2] * 8 = (7/2) * 8 = 7*8/2 = N(N+1)/2 Either way, the formula comes out the same. This is not a proof, by the way. It just "motivates" the formula in an intuitive sense. From here, the finite math books usually say something about how the "motivated" reader will chase through the inductive proof. Here’s where Randall comes in. As the arbiter of correctness, he will now be compelled to either produce the inductive proof for this formula or tell us that it is too trivial to bother with it here. Since we are in the midst of the Holiday season, let’s hope he does the latter.<wink> I am sure that Randall has seen not only seen this proof, but done it as part of his training. ***** OK! We’re back! ***** Jegs, if you solved this problem without peeking, I’d agree that you showed some ingenuity. You must be fairly clever to have cooked up a solution for it without the general formula. So much for "not being good at math." Hurray for you! That’s good work. Sorry to hear that it took two tries to get through the math course, though. A good tutor could have made a world of difference there. Usually, there is more than one way to explain the concepts that the students are expected to master in a math course. Teachers tend to try to save time by using their own favorite approach in introducing each chapter or section. Often, just hearing a second explanation from a slightly different angle can open the way to a clearer understanding. I have done a lot of tutoring in the past, and I am always amazed to see how little I have to do to get someone back on track if the problem is addressed early on in the year. "Getting it" in math depends on making the connection from the ideas that are familiar to the new ideas that build on those earlier building blocks. The more alternative paths you can build from the old to the new, the more likely it is that you’ll get it. Any adults looking for volunteer opportunities? How about registering with your local TSA chapter and/or your school district as a volunteer tutor for math or reading? They’d probably be willing to give you some training. I know they’d be willing to give you some students.<g> Guess I’m always looking for ways to get people involved in service to others. If any of you have an interest in this area, I urge you to pursue it. Tutoring is "one of the toughest jobs you’ll ever love." Being there when the lights come on in a kid’s eyes is one of the best experiences I know of in this world. I’ll warn you that tutoring is hard work that requires intense concentration and good listening skills, but I just don’t know of anything more satisfying. With me, it is a labor of love, a way to draw out the potential that no one knew was there. It is one more way to wipe out that word "can’t" that everyone keeps using to club us over the head. Any time we can eliminate a "can’t" in someone’s life, we have opened up possibilities for a richer, fuller life. That’s like setting a person’s spirit free. Show some kindness this weekend. Freedom and love have a peculiar physics all their own. The more you give, the more you’ll have for yourself. Be free, Chuck Jegs112 wrote: > I almost forgot, one day I was paging through an Omni magazine (is it still > published?) and there was an ad from MENSA, with the question: > "if you added one to 100 as 1+2+3…+99+100) what would your total be?" > For someone with advanced algebra, no problem, but for some guy who hadn’t > taken math in several years, it was an interesting puzzle.
[snip] – Hide quoted text — Show quoted text -> by the way the short version is sigma: (x * (x-1))/2 > but I didn’t know that until my freind told me that. I don’t remember my > version anymore, but a math professor freind of mine said that it was quite > ingenious. > By the way, that still didn’t remedy the fact that I still needed to take my > math classes twice to pass.
Response:
In article <19971212021000.VAA07…@ladder01.news.aol.com>, jegs…@aol.com (Jegs112) wrote: – Hide quoted text — Show quoted text -> I almost forgot, one day I was paging through an Omni magazine (is it still > published?) and there was an ad from MENSA, with the question: > "if you added one to 100 as 1+2+3…+99+100) what would your total be?" > For someone with advanced algebra, no problem, but for some guy who hadn’t > taken math in several years, it was an interesting puzzle. > Well, I finally figured it out and showed my neighbor across the hall (I was in > college) to look at it. Anyway, he couldn’t figure it out, although the > expression worked, he finally reduced it to what his math book showed. (he was > a chemistry major) I asked him how he reduced it. After 45 min. of dabbling > with my expression, he finally said that he just pulled the reduced expression > out of his math book. > by the way the short version is sigma: (x * (x-1))/2 > but I didn’t know that until my freind told me that. > I don’t remember my version anymore, but a math professor freind of mine said > that it was quite ingenious. > By the way, that still didn’t remedy the fact that I still needed to take my > math classes twice to pass. > .
The easy way to remember the formula is to realize that if you add exactly the same sequence of N numbers to the given sequence, backwards, term by term, you get N numbers equal to N+1, but since you added the sequence to itself you actually have twice as much as you really want so you divide by 2. I always heard that some famous mathematician did this as a kid in school when his teacher gave him something that the teacher thought would keep him occupied for a long time and thus out of the teacher’s hair. The teacher failed. ——————-==== Posted via Deja News ====———————– http://www.dejanews.com/ Search, Read, Post to Usenet
Response:
I almost forgot, one day I was paging through an Omni magazine (is it still published?) and there was an ad from MENSA, with the question: "if you added one to 100 as 1+2+3…+99+100) what would your total be?" For someone with advanced algebra, no problem, but for some guy who hadn’t taken math in several years, it was an interesting puzzle. Well, I finally figured it out and showed my neighbor across the hall (I was in college) to look at it. Anyway, he couldn’t figure it out, although the expression worked, he finally reduced it to what his math book showed. (he was a chemistry major) I asked him how he reduced it. After 45 min. of dabbling with my expression, he finally said that he just pulled the reduced expression out of his math book. by the way the short version is sigma: (x * (x-1))/2 but I didn’t know that until my freind told me that. I don’t remember my version anymore, but a math professor freind of mine said that it was quite ingenious. By the way, that still didn’t remedy the fact that I still needed to take my math classes twice to pass. .
