1. Some natural numbers can be expressed as a difference of two squares, but others cannot. For example, 12=4^2 - 2^2, but it is impossible to write 10 as a difference of two squares. Find a way to determine whether or not a given natural number can be expressed as the difference of two perfect squares.
2. In rectangle ABCD, M is the midpoint of side AD and N is the midpoint of side DC. Segments AN and CM intersect at E
a) Prove that <AEM=<MBN
b)Determine how the angles in part a are related to the length: width ratio of the rectangle.
3. A sequence is defined by the equation below:
a) Prove that every term of the sequence is less than 3.
b) Prove that every term of the sequence is greater than the preceding term.
Any help you can give me on any/all of these questions would be great, thanks.
Now i'm one to always be off by some technicality, so use these answers as a general outline and fix it up to your liking.
Let the sequence be defined recursively as , for all integers
We show that for all integers
We proceed by induction.
Let P(n): " for all integers "
For P(1) we have , so P(1) is true.
Assume P(k) is true for some integer
Since P(k) is true, we have
So we have , which is P(k + 1)
Thus the induction proof is complete, and we have for all integers
Let be the sequence defined above. We show that for all integersb) Prove that every term of the sequence is greater than the preceding term.
We proceed by induction.
Let P(n): " for all integers "
We have for P(1): . So P(1) is true.
Assume P(k) is true for some integer
Then we have:
, which is P(k + 1)
Thus the induction proof is complete and we have for all integers
Hello, omgmath!
How about an inductive proof?3. A sequence is defined by: .
a) Prove that every term of the sequence is less than 3.
b) Prove that every term of the sequence is greater than the preceding term.
(a) for all
Verify . . . true
Assume is true: .
Multiply both sides by 2: .
Add 1 to both sides: .
Take the square root of both sides: .
The left side is:
The right side is:
Therefore: .
. . The inductive proof is complete.
Part (b) is proved in a similar manner.
[Edit: Jhevon beat me to it . . .]
This is a number theory question.
Say,
A possible factorization of is
And hence it is reasonable to search for solutions to:
The solution is given by .
Now the thing that we need is for to be integers. And that certainly happens if is an odd positive integer.
What happens if it is even? The possibilities is that has one of the four forms: . Note we covered the cases . Let us do . Then since it is divisible by 4 we can write:
And what happens if . Well, the is impossible because a square modulo 4 must be either 1 or 3. Hence . Not 2! Hence such a representation is impossible.
I have an ugly way of doing it, but I think it works.
Look at picture below. We want to show the blue angles are equal to eachother.
In coordinate geometry you perhaps learn the following theorem, let be two lines (non-vertical). And let has bigger angle measurement then with the x-axis. Then if is the angle between them we have:
where is the slope of and is the slope of .
Here is how we use it.
Let be the angle <AEM. And be the angle <MBN. We will show that and deduce .
Now we find and by using the theorem above.
Let me do and leave for you to do.
In the picture is formed by lines AN and CM with side CM creating a bigger angle than AN with x-axis (in this case x-axis is AD).
Thus,
.
To compute the slope we just find the ratio between the the height and width.
Thus,
This answers the second part of the question on how to compute the angle in terms of the sides.
Now do the same thing with .
And show they give the same value.
Suppose integer X can be written as the difference of two squares, then:
X = a^2-b^2 = (a-b)(a+b)
then if we suppose a>b>0, then X must have two non-zero factors u, v,
u!=v, where
u=a+b
v=a-b
adding these we see that u+v=2a, and subtracting these we see that
u-v=2b.
Now suppose we are given any non zero u and v, u>v, such that their sum is even (then their difference is even as well), we can write:
a=(u+v)/2
and:
b=(u-v)/2
then uv=(a+b)(a-b).
So we see that for any X to be written as the difference of two squares
it is necessary and sufficient that X have two distinct factors whoes sum
is even.
RonL
Hello, omgmath!
I have an even uglier solution for #2 . . .
Let2. In rectangle ABCD, M is the midpoint of side AD and N is the midpoint of side DC.
. . Segments AN and CM intersect at E
a) Prove that <AEM=<MBN
b) Determine how the angles in part a are related to the length: width ratio of the rectangle.
Let
Let
In
. . Hence: .
We have: .
. . we have: .
. . which simplifies to: . . [1]
Here's the uglier part . . .
In right triangle
In right triangle
In right triangle
. . which simplifies to: .
Since:
. . Then: .
And we have: . . [2]
Therefore, from [1] and [2]: .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Let
We have: .
Therefore: .