That’s way too much work
" prove that 5^n+9^n+2 is divisible by 4 for all natural numbers n" is the same thing as saying
"prove 5^n+9^n+2 = 0 mod 4
Since 5 = 9 = 1 mod 4, we have 5^n+9^n+2 = 1^n + 1^n + 2 = 4 = 0 mod 4.
I wouldn’t suggest doing too much work along my lines. Actually I would consider the sentence I wrote pretty much a complete proof and would be pretty pissed if someone made me write out any more steps lol. Maybe (4+1)^n=4(integer)+1 is as much further as I’d go.
I don’t know. To me it sounds more like a sketch of a proof than a proof, or even something you would read from a problem list at the end of a section. If you didn’t feel like expanding the binomial theorem, your could at least say something like: Notice that for n>0, 4^n*1^n(n,k) is a multiple of 4. Just giving the reader a sketch as opposed to a full demonstration doesn’t really prove what you are trying to show.
working on summation problems got through the first few but i don’t know how i would separate the i
4
Σ 1/(i^2+1)
i=0
What do you mean by separate the i? Just do the sum.
1/(0^2+1)+1/(1^2+1)+…+1/(4^2+1)=?
I’ll be frequenting this thread a lot often, as I’m taking a shit-ton of math courses soon for my major.
Might have questions for you guys.
You don’t separate the i at all. You substitute the number for the i’th term for the i. Your sum goes from 0 to 4, so you are going to do 5 substitutions in total.
As a simplified example, rather than writing something like
2 + 4 + 6 + 8 + 10
you can write it as
5
Σ 2i
i=1
these are both the same things:
Of course, you can factor the 2 our of the sum to get
5
2 Σ i
i = 1
space
The little clock/gear thingie to the right edits post.
Also, I have a friend that didn’t do super hot in Pre Calc (he got a D). Would you guys recommend the prerequisite of Trig/College Algebra to Calc 1 or Trig/Pre-Calc? Seems like the general consensus around the web is that Pre-Calc prepares you for Calc 1 better than Trig/CA.
Good question. It really depends what pre-calc is, and I’m not sure I even know. From what I’ve seen, it’s basically trig, logarithms and exponents, and then basic limit concepts. So it will be more helpful in the sense that it actually includes some calculus.
However, it’s useless to learn that stuff if you don’t really know algebra in the first place, so I don’t know. My experience is that almost everyone needs a good review of algebra, and unfortunately most of them think they know it better than they do. If you can get your hands on a decent algebra book and do almost all of the problems, you’re probably ready for pre-calc.
To kind of expand on what BBQ is saying, taking trig and precalc without a solid knolwedge of algebra is like signing up to run a marathon when you’re still having trouble walking. You will be expected to algebraically manipulate expressions like it’s nothing. One thing that I’ve seen constantly for instance, is when you take trig and you have to prove trig identities. Nobody has a clue what to do with those because there isn’t an example in the book that covers it because you have to use algebra, substitutions, and previous identities to arrive at the desired conclusion. You will be expected to know things like binomial expansion, factoring, completing the square, and even when to make an appropriate algebraic substitution, all in order to compute the limits of functions. That’s why if you can’t do algebra like it’s nothing, you most likely will not succeed in precalc and especially not in calc. So, if you were to put all your eggs in one basket, make sure it’s algebra.
This week I realized how much I hate topology. I guess in math you can’t like everything.
what don’t you like about it?
That I suck at it. Proofs like this one , the proof itself is really easy to understand but I would have never thought about it (The exercise in the textbook only said “the arithmetic progressions generate a topology, use it to proof the inifinitude of primes”).
Don’t get too discouraged. The whole point of this course is to think in terms of sets, which you no doubt have been working on all semester. They used arithmetic progressions because it is something that is supposed to be familiar to students taking an advanced math course. What may not be familair is “the infinitude of primes,” which is a classic in number theory where they use a proof by contradiction to demonstrate it. Iirc, it goes something like this:
Suppose there are only a finite number of primes, then you can create a number whoses canonical decomposition (i.e. factored form) is composed of each prime exactly once. Since there are infintely many numbers, it means there exists a number that is one more than this number. But since none of the primes will divide the number, it means that this new number is also a prime) since both 1 and the number itself are the only numbers that divide it), contradicting our original assumption.
Typically, if you’re trying to prove there are an unlimited number of something, contradiction is a good first idea.
It’s hard to know what to do with something like that if you’re new to topology, so I wouldn’t feel too bad about it. I actually don’t know much topology myself, besides the basics up to like Baire Category Theorem or something (and I forget what that says too, something about “zooming in” and always finding another point seems to sound familiar lol).
But I do remember the definition of topology, so my thinking was first to let the open sets be the arithmetic progressions, as the hint suggests, and then verify that you get a topology. You actually don’t because the union of {…,-5,0,5,…} and {…,-2,0,2,…} is not an arithmetic progression. Then you just try defining open sets as unions of arithmetic progressions to get around this and that works.
I don’t know what he was thinking when he came up with the proof, but he most likely started from the last line, by considering unions* (since you don’t take infinite intersections but you can take infinite unions, it seems that unions have to be the key to the proof, if there is one). And what would you take unions of? It seems pretty natural to at least try the sets {…,-p,0,p,…} for all primes p. This union is Z/{-1,1}, and then I guess you just have to play around with the basic definitions of open/closed, etc… I won’t pretend that I carried this through, but I got that far and convinced myself that maybe with some fumbling around and a topology book for reference I could do the rest.
The point is that nobody just reasons out a proof like it’s presented on that wikipedia page or in most textbook, and I think a lot of students think they’re not any good at math because they have trouble following a slick proof or they can’t come up with it in the order it’s written within an hour. It doesn’t work that way.
*Actually, even this step for him wouldn’t be the beginning. He didn’t have the statement in front of him to prove since he’s the one who came up with it, so his start was probably just playing around with progressions of primes. Then maybe he had an insight because he had just learned topology and it was fresh on his mind.
Anybody know of any good low cost/free resources on differential equations? I never got a chance to take it in college. I was hoping Coursera would have something, but no luck.
I did find this http://tutorial.math.lamar.edu/Classes/DE/DE.aspx
It’s basically a collection of someone’s notes. Seems like a good starting point, I think, but I’d like more.
How accurate is this statement?
“Most people get through Calculus I, Limits and Derivatives, without too much difficulty but Calculus II, Integration and Infinite Series, tends to be a merciless weeder course that ends the careers of many aspiring engineers. I think it is because unlike all previous courses, you can not just memorize a load of formulas, functions and derivations and stick in the numbers.”
I need Calc 3 and Liner Algebra to get my CPUSci Math Degree. Currently in Calc 1.
maybe it depends on your instructor but I thought Calc 1 and 2 (as well as the first couple Physics w/Calculus courses) were relatively equal weeder courses.
btw CPU is not an abbreviation for computer
For me all the calcs were easy, but then again I finished them in high school. People complained more about partial differential equations but what really knocked out a lot of college students were the computer science algorithms courses.