not sure of solution method

Fuzell Wood

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i failed to use LaTeX lang so i better attach a photo of the task

20190520_140553.jpg
 
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What is the context of the question? You imply that you think it is to be solved by a differential equation; what reason do you have for that assumption? Is a formal proof required, or would a qualitative answer based on a geometrical picture be sufficient?
 
What is the context of the question? You imply that you think it is to be solved by a differential equation; what reason do you have for that assumption? Is a formal proof required, or would a qualitative answer based on a geometrical picture be sufficient?
as i have said i don't know. i suppose that this limit could be solved by integral. If you know, tell me please how i can solve it or what kind of article i should read and search for getting solution or which thread it will be more appropriate to post the question in
 
Please answer my question: What is the context? If it is for a class, what class is it, and what topics are you learning? What techniques have you been taught that it might be intended to test? Have you seen any other problems involving lattice points (integer coordinate pairs)?

And did you consider what I suggested about just sketching what it is asking?
 
I havent been taught any techniques,it is not for a class, this is the sample of admission test to university.
Everything is on the photo, the task and what it is asking, i have no more than that and i have never faced this sort of problem
 
That is important information. When you read (I assume) our submission guidelines, you should have seen this:

1. From where does your question come?
It really helps to know why you're working on math or what math course you're taking. There are many ways to find the same answers. We'd like to discuss the method that you're learning and to explain at your level of study.​

Now we know that this should be solvable either by commonly taught pre-university math, or by insights that they expect to distinguish excellent candidates from the ordinary. This appears to be the latter; so (a) you should not be too concerned that you don't see an immediate method, and (b) you will have to let yourself imagine anything you have learned that might help. It will not be a matter of applying some routine formula!

Again, try drawing a picture. What region of the plane does each inequality represent? What familiar geometrical idea might the limit represent?
 
That is important information. When you read (I assume) our submission guidelines, you should have seen this:

1. From where does your question come?
It really helps to know why you're working on math or what math course you're taking. There are many ways to find the same answers. We'd like to discuss the method that you're learning and to explain at your level of study.​

Now we know that this should be solvable either by commonly taught pre-university math, or by insights that they expect to distinguish excellent candidates from the ordinary. This appears to be the latter; so (a) you should not be too concerned that you don't see an immediate method, and (b) you will have to let yourself imagine anything you have learned that might help. It will not be a matter of applying some routine formula!

Again, try drawing a picture. What region of the plane does each inequality represent? What familiar geometrical idea might the limit represent?

i did read the guidelines.Okay, but what is "n" here? If an is the number of pairs so what is n?
 
Now you're asking appropriate specific questions. Thanks. We can talk about the meaning of a question more easily than about what you in particular can expect to do for your final solution.

What they are doing is forming a sequence, where n is first 1, then 2, then 3, and so on. For example, [MATH]a_1[/MATH] is the number of coordinate pairs (that is, points) [MATH](s, t)[/MATH] inside the region (for n=1) [MATH]s^2 + t^2 \le 1[/MATH], and outside of the region [MATH]|s| + |t| < 1[/MATH]. Sketch that region, count points to find [MATH]a_1[/MATH], and evaluate [MATH]\frac{a_n}{n^2}[/MATH]. Then do the same for n=2, and maybe a couple more (n = 5 maybe interesting). Then sketch the region (without trying to show all the points) for some larger value of n, say n = 100. Then think about what will happen for really large numbers.

Hint: think about areas.
 
No, n is any positive number. They appear to be thinking of n as an integer, so they are talking about a sequence: the first number in the sequence, called [MATH]a_1[/MATH], is the number of points in the region defined for n=1, and so on. The number n is the index of the sequence, identifying a member of the sequence (first, second, ..., nth).

Let's back up. The problem mentions a limit; do you know what limits are? It implicitly talks about a sequence; do you know what a sequence is? Have you seen notations like [MATH]a_n[/MATH]? If not, then ignore this question - it's not for you! Or else, look up sequences in a textbook.
 
[MATH]a_n[/MATH] is a number, the number of co-ordinate pairs specified by n.

[MATH]n[/MATH] is also a number, in this case, the index identifying which a in the sequence of a's being discussed. That is you have an infinite list of numbers

[MATH]a_1,\ a_2,\ a_3,\ ...[/MATH]
What makes this a little weird is that [MATH]a_n = f(n)[/MATH]
so you could think of this as the sequence

[MATH]f(1),\ f(2),\ f(3),\ ...[/MATH]
and you are being asked to find the limit

[MATH]\lim_{n \rightarrow \infty} \dfrac{f(n)}{n^2}.[/MATH]
Changing topics, I would not start by thinking about integrals because the question is about integers. I am not saying that you cannot sometimes use calculus to attack problems involving integers, but calculus directly applies to continuous functions.

Finally, Dr. P has asked you to think about computing [MATH]a_1.[/MATH]
That is truly excellent advice.

Given that s and t are integers, what might they be if

[MATH]s^2 + t^2 \le 1^2 = 1.[/MATH]
For example, s = 0 and t = 0 [MATH]\implies s^2 + t^2 \le 1.[/MATH]
How many of those pairs also satisfy

[MATH]|s| + |t| \ge 1.[/MATH]
Having answered those two questions can you say what [MATH]a_1 =?[/MATH]
The purpose of this problem is to see if you can figure out what the problem is even asking about and then see how to solve a problem that is not of a routine type.
 
No, n is any positive number. They appear to be thinking of n as an integer, so they are talking about a sequence: the first number in the sequence, called [MATH]a_1[/MATH], is the number of points in the region defined for n=1, and so on. The number n is the index of the sequence, identifying a member of the sequence (first, second, ..., nth).

Let's back up. The problem mentions a limit; do you know what limits are? It implicitly talks about a sequence; do you know what a sequence is? Have you seen notations like [MATH]a_n[/MATH]? If not, then ignore this question - it's not for you! Or else, look up sequences in a textbook.
yes,i do know this. you said that i have to sketch regions, but i'm just trying to understand what is what on the graph and how i can do it
 
[MATH]a_n[/MATH] is a number, the number of co-ordinate pairs specified by n.

[MATH]n[/MATH] is also a number, in this case, the index identifying which a in the sequence of a's being discussed. That is you have an infinite list of numbers

[MATH]a_1,\ a_2,\ a_3,\ ...[/MATH]
What makes this a little weird is that [MATH]a_n = f(n)[/MATH]
so you could think of this as the sequence

[MATH]f(1),\ f(2),\ f(3),\ ...[/MATH]
and you are being asked to find the limit

[MATH]\lim_{n \rightarrow \infty} \dfrac{f(n)}{n^2}.[/MATH]
Changing topics, I would not start by thinking about integrals because the question is about integers. I am not saying that you cannot sometimes use calculus to attack problems involving integers, but calculus directly applies to continuous functions.

Finally, Dr. P has asked you to think about computing [MATH]a_1.[/MATH]
That is truly excellent advice.

Given that s and t are integers, what might they be if

[MATH]s^2 + t^2 \le 1^2 = 1.[/MATH]
For example, s = 0 and t = 0 [MATH]\implies s^2 + t^2 \le 1.[/MATH]
How many of those pairs also satisfy

[MATH]|s| + |t| \ge 1.[/MATH]
Having answered those two questions can you say what [MATH]a_1 =?[/MATH]
The purpose of this problem is to see if you can figure out what the problem is even asking about and then see how to solve a problem that is not of a routine type.

is it going to be an arithmetic progression?
 
No, it will not be anything simple you can write an equation for.

What does the graph of x^2 + y^2 = 1 look like?

What does the graph of |x| + |y| = 1 look like?

Please try this. I don't believe you can think about the problem at all without doing this.
 
i have drawn the region for a1 and it looks like a rhombus incribed in a circle. Am is supposed to find the area of a region outside of this rhombus but inside the circle?
S=piR^2-a^2 that is pi-2 for a1
however, i cannot understand why i solve the limit by finding this area
 
No, n is any positive number. They appear to be thinking of n as an integer, so they are talking about a sequence: the first number in the sequence, called [MATH]a_1[/MATH], is the number of points in the region defined for n=1, and so on. The number n is the index of the sequence, identifying a member of the sequence (first, second, ..., nth).

Let's back up. The problem mentions a limit; do you know what limits are? It implicitly talks about a sequence; do you know what a sequence is? Have you seen notations like [MATH]a_n[/MATH]? If not, then ignore this question - it's not for you! Or else, look up sequences in a textbook.
i know i'm not that(totally) good at maths, so that's why i'm here and asking for help as neither groupmates nor teachers are not able to help me with these tasks. But i'm really willing to work this out, understand and develop my skills in order to manage to solve problems where "routine formulas" cannot be used.
I REALLY appreciate your help and patience. Thank you very much.
 
i have drawn the region for a1 and it looks like a rhombus inscribed in a circle. Am is supposed to find the area of a region outside of this rhombus but inside the circle?
S=piR^2-a^2 that is pi-2 for a1
however, i cannot understand why i solve the limit by finding this area
You've got the idea. That rhombus is a square, and you've found the area correctly, though you didn't write things quite right. The radius of the circle is [MATH]n[/MATH], and the diagonal of the square is [MATH]2n[/MATH], so the area is [MATH]\pi n^2 - 2 n^2[/MATH], and the ratio [MATH]\frac{A}{n^2}[/MATH] is [MATH]\pi - 2[/MATH], for any [MATH]n[/MATH].

To see why this would give the limit you are asked for, consider that apart from regions near the boundary, each lattice point (the pairs you're counting) can be considered the center of a square inside the region. As the figure gets larger, a smaller proportion of the area is near boundaries, so the number of points approaches the area. That is, [MATH]a_n[/MATH] approaches [MATH]A[/MATH], and the ratio [MATH]\frac{a_n}{n^2}[/MATH] approaches [MATH]\pi - 2[/MATH].

A thorough answer would have to consider how the boundaries affect the count, and confirm that the ratio approaches exactly this limit. But I think this is sufficient, as a university would not expect incoming students to be good at college-level proofs.
 
i don't understand the part where you say that each lattice point can be considered as the centre of a square. Can you sketch and show it on the graph?
 
Here is a picture, with n=6, where the first quadrant shows the lattice points and the second quadrant shows the squares centered on them, whose areas approximate that of the green region:
FMH116225.png
 
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