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Thread: Please help with exam prep! - Proofs

  1. #1
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    Please help with exam prep! - Proofs

    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.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by omgmath View Post

    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.
    The sequence equation is missing
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  3. #3
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    Sorry I forgot the last equation, here it is:
    Attached Thumbnails Attached Thumbnails Please help with exam prep! - Proofs-receqn.bmp  
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by omgmath View Post

    3. A sequence is defined by the equation below:

    $\displaystyle t_1 = 1$, $\displaystyle t_{n + 1} = \sqrt {2 t_n + 1}$

    a) Prove that every term of the sequence is less than 3.
    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 $\displaystyle \left( t_n \right) $ be defined recursively as $\displaystyle t_1 = 1$, $\displaystyle t_{n + 1} = \sqrt { 2 t_n + 1}$ for all integers $\displaystyle n \geq 1$

    We show that $\displaystyle t_n < 3$ for all integers $\displaystyle n \geq 1$

    We proceed by induction.

    Let P(n): "$\displaystyle t_n < 3$ for all integers $\displaystyle n \geq 1$"

    For P(1) we have $\displaystyle t_1 = 1 < 3$, so P(1) is true.

    Assume P(k) is true for some integer $\displaystyle k > 1$

    Since P(k) is true, we have

    $\displaystyle t_k < 3$

    $\displaystyle \sqrt {2 t_{k - 1} + 1} < 3$

    $\displaystyle \Rightarrow 2 \sqrt {2 t_{k - 1} + 1} < 2 \cdot 3 = 6$

    $\displaystyle \Rightarrow 2 \sqrt {2 t_{k - 1} + 1} + 1 < 6 + 1 = 7$

    $\displaystyle \Rightarrow \sqrt { 2 \sqrt {2 t_{k - 1} + 1} + 1} < \sqrt {7}$

    $\displaystyle \Rightarrow \sqrt {2 t_k + 1} < \sqrt {7}$

    $\displaystyle \Rightarrow t_{k + 1} < \sqrt {7} < 3$

    So we have $\displaystyle t_{k + 1} < 3$, which is P(k + 1)

    Thus the induction proof is complete, and we have $\displaystyle t_n < 3$ for all integers $\displaystyle n \geq 1$


    b) Prove that every term of the sequence is greater than the preceding term.
    Let $\displaystyle \left( t_n \right)$ be the sequence defined above. We show that $\displaystyle t_{n + 1} > t_n$ for all integers $\displaystyle n \geq 1$

    We proceed by induction.

    Let P(n): "$\displaystyle t_{n + 1} > t_n$ for all integers $\displaystyle n \geq 1$"

    We have for P(1): $\displaystyle t_{1 + 1} = t_2 = \sqrt {3} > 1 = t_1 $. So P(1) is true.

    Assume P(k) is true for some integer $\displaystyle k \geq 1$

    Then we have:

    $\displaystyle t_{k + 1} > t_k$

    $\displaystyle \Rightarrow 2 t_{k + 1} > 2 t_k$

    $\displaystyle \Rightarrow 2 t_{k + 1} + 1 > 2 t_k + 1$

    $\displaystyle \Rightarrow \sqrt {2 t_{k + 1} + 1} > \sqrt {2 t_k + 1}$

    $\displaystyle \Rightarrow t_{k + 2} > t_{k + 1}$, which is P(k + 1)

    Thus the induction proof is complete and we have $\displaystyle t_{n + 1} > t_n$ for all integers $\displaystyle n \geq 1$
    Last edited by Jhevon; Jun 17th 2007 at 06:33 PM.
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  5. #5
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    Hello, omgmath!

    3. A sequence is defined by: .$\displaystyle t_1 = 1,\;t_{n+1} \:=\:\sqrt{2t_n + 1}$

    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.
    How about an inductive proof?

    (a) $\displaystyle S(n)\!: \;t_n < 3$ for all $\displaystyle n.$

    Verify $\displaystyle S(1)\!:\;t_1 \,=\,1 \:<\:3$ . . . true


    Assume $\displaystyle S(k)$ is true: .$\displaystyle t_k \:< \:3$

    Multiply both sides by 2: .$\displaystyle 2\!\cdot\!t_k \:<\:6$

    Add 1 to both sides: .$\displaystyle 2\!\cdot\!t_k + 1 \:<\:7$

    Take the square root of both sides: .$\displaystyle \sqrt{2\!\cdot\!t_k + 1} \:< \:\sqrt{7}$


    The left side is: $\displaystyle t_{k+1}$
    The right side is: $\displaystyle \sqrt{7} \:=\:2.645... \:<\:3$

    Therefore: .$\displaystyle t_{k+1} \:<\:3$
    . . The inductive proof is complete.

    Part (b) is proved in a similar manner.


    [Edit: Jhevon beat me to it . . .]
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Soroban View Post
    [Edit: Jhevon beat me to it . . .]
    you were a lot smoother than me though. i went back to defining $\displaystyle t_k$ in terms of $\displaystyle t_{k - 1} $, which was totally unnecessary and unaesthetic.

    you can go after the other two if you want
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    Thanks guys, any ideas on the first 2?
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    Quote Originally Posted by omgmath View Post
    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.
    This is a number theory question.

    Say,
    $\displaystyle n = a^2 - b^2 = (a+b)(a-b)$
    A possible factorization of $\displaystyle n$ is $\displaystyle n\cdot 1$

    And hence it is reasonable to search for solutions to:
    $\displaystyle a+b=n$
    $\displaystyle a-b=1$

    The solution is given by $\displaystyle (a,b) = \left( \frac{n+1}{2}, \frac{n-1}{2} \right)$.

    Now the thing that we need is for $\displaystyle a,b$ to be integers. And that certainly happens if $\displaystyle n$ is an odd positive integer.

    What happens if it is even? The possibilities is that $\displaystyle n$ has one of the four forms: $\displaystyle 4k,4k+1,4k+2,4k+3$. Note we covered the cases $\displaystyle 4k+1,4k+3$. Let us do $\displaystyle n=4k$. Then since it is divisible by 4 we can write:
    $\displaystyle n = \left( \frac{n}{4} + 1 \right)^2 - \left( \frac{n}{4} - 1 \right)^2 $

    And what happens if $\displaystyle n=4k+2$. Well, the is impossible because a square modulo 4 must be either 1 or 3. Hence $\displaystyle a^2 - b^2 = 0,1,3 (\bmod 4)$. Not 2! Hence such a representation is impossible.
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  9. #9
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    Is there any way you could explain that to me a bit better, I have no clue how you got the solution for (a,b), and I dont understand what you were getting at when you said they need to be integers if n i odd...Im basically completely lost
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    Quote Originally Posted by omgmath View Post
    Is there any way you could explain that to me a bit better, I have no clue how you got the solution for (a,b), and I dont understand what you were getting at when you said they need to be integers if n i odd...Im basically completely lost
    Add the two equations to eliminate b.

    Subtract the two equation to eliminate a.

    Do you know how to solve simulatenous equations?
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    Quote Originally Posted by omgmath View Post
    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.
    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 $\displaystyle l_2,l_1$ be two lines (non-vertical). And let $\displaystyle l_2$ has bigger angle measurement then $\displaystyle l_1$ with the x-axis. Then if $\displaystyle \theta$ is the angle between them we have:
    $\displaystyle \tan \theta = \frac{m_2 - m_1}{1+m_2m_1}$ where $\displaystyle m_2$ is the slope of $\displaystyle l_2$ and $\displaystyle m_1$ is the slope of $\displaystyle l_1$.

    Here is how we use it.

    Let $\displaystyle \alpha$ be the angle <AEM. And $\displaystyle \beta$ be the angle <MBN. We will show that $\displaystyle \tan \alpha = \tan \beta$ and deduce $\displaystyle \alpha = \beta$.

    Now we find $\displaystyle \tan \alpha$ and $\displaystyle \tan \beta$ by using the theorem above.

    Let me do $\displaystyle \alpha$ and leave $\displaystyle \beta$ for you to do.

    In the picture $\displaystyle \alpha$ 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,
    $\displaystyle \tan \alpha = \frac{m_{CM} - m_{AN}}{1 + m_{CM}\cdot m_{AN}}$.

    To compute the slope we just find the ratio between the the height and width.

    $\displaystyle m_{CM} = \frac{CD}{DM} = \frac{2DN}{DM}$

    $\displaystyle m_{AN} = \frac{DN}{AD} = \frac{DN}{2DM}$

    Thus,
    $\displaystyle \tan \alpha = \frac{\frac{2DN}{DM} - \frac{DN}{2DM}}{1+ \frac{2DN}{DM}\cdot \frac{DN}{2DM}} = \frac{\frac{3DN}{2DM}}{\frac{DM^2+DN^2}{DM^2}} = \frac{3DN\cdot DM}{2MN^2}$

    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 $\displaystyle \tan \beta$.
    And show they give the same value.
    Attached Thumbnails Attached Thumbnails Please help with exam prep! - Proofs-picture17.gif  
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  12. #12
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    Quote Originally Posted by omgmath View Post
    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.

    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
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    Hello, omgmath!

    I have an even uglier solution for #2 . . .


    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.
    Let $\displaystyle L = AD,\;W = CD$

    Let $\displaystyle \angle CMD = \alpha\quad\Rightarrow\quad\angle MCD \,=\, 90 -\alpha$

    Let $\displaystyle \angle NAD = \beta\quad\Rightarrow\quad\angle AND = 90-\beta\quad\Rightarrow\quad\angle ANC = 90 + \beta$

    In $\displaystyle \Delta CEN:\;\angle CEN \;=\;180^o - (90 - \alpha) - (90 + \beta) \;=\;\alpha - \beta$

    . . Hence: .$\displaystyle \angle AEM \;=\;\alpha - \beta$

    We have: .$\displaystyle \tan(\angle AEM) \;=\;\tan(\alpha - \beta) \;=\;\frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\cdot\tan\beta} $

    $\displaystyle \text{Since: }\tan\alpha \:=\:\frac{W}{\frac{L}{2}} \:=\:\frac{2W}{L}\;\text{ and }\;\tan\beta \:=\:\frac{\frac{W}{2}}{L} \:=\:\frac{W}{2L}$

    . . we have: .$\displaystyle \tan(\angle AEM) \;=\;\frac{\frac{2W}{L} - \frac{W}{2L}}{1 + \left(\frac{2W}{L}\right)\left(\frac{W}{2L}\right) }$

    . . which simplifies to: .$\displaystyle \boxed{\tan(\angle AEM) \:=\:\frac{3LW}{2(L^2+W^2)}}$ . [1]


    Here's the uglier part . . .

    In right triangle $\displaystyle BAM\!:\;BM^2 \:=\:\left(\frac{L}{2}\right)\!^2 + W^2\quad\Rightarrow\quad BM \:=\:\frac{\sqrt{L^2+4W^2}}{2}$
    In right triangle $\displaystyle BCN\!:\;BN^2 \:=\:L^2 + \left(\frac{W}{2}\right)\!^2\quad\Rightarrow\quad BN \:=\:\frac{\sqrt{4L^2+W^2}}{2}$
    In right triangle $\displaystyle NDM\!:\;MN^2 \:=\:\left(\frac{L}{2}\right)\!^2 + \left(\frac{W}{2}\right)\!^2\quad\Rightarrow\quad MN \:=\:\frac{\sqrt{L^2+W^2}}{2}$

    $\displaystyle \text{Then: }\;\cos(\angle MBN) \:=\:\frac{BM^2 + BN^2 - MN^2}{2(BM)(BN)} \:=\:\frac{\frac{L^2+4W^2}{4} +\frac{4l^2+W^2}{4} - \frac{L^2+W^2}{4}}{2\left(\frac{\sqrt{L^2+4W^2}}{2 }\right)\left(\frac{\sqrt{4L^2+W^2}}{2}\right)}$

    . . which simplifies to: .$\displaystyle \cos(\angle MBN) \;=\;\frac{2(L^2+W^2)}{\sqrt{L^2+4W^2}\sqrt{4L^2+W ^2}} \;=\;\frac{adj}{hyp} $

    Since: $\displaystyle opp^2 \:=\:hyp^2 - adj^2\quad\Rightarrow\quad opp^2 \;=\;\left[\sqrt{L^2+4W^2}\sqrt{4L^2+W^2}\right]^2 - \left[2(L^2+W^2)\right]^2$

    . . Then: .$\displaystyle opp^2 \:=\:9L^2W^2\quad\Rightarrow\quad opp \:=\:3LW$

    And we have: .$\displaystyle \boxed{\tan(\angle MBN) \;=\;\frac{3LW}{2(L^2+W^2)}}$ . [2]


    Therefore, from [1] and [2]: .$\displaystyle \angle AEM \:=\:\angle MBN$

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Let $\displaystyle r \,=\,\frac{L}{W}$
    We have: .$\displaystyle \tan\theta \;=\;\frac{3LW}{2(L^2+W^2)}$
    $\displaystyle \text{Divide top and bottom by }W^2\!:\;\;\tan\theta \;=\;\frac{3\left(\frac{L}{W}\right)}{2\left[\left(\frac{L}{W}\right)^2 + 1\right]} \;=\;\frac{3r}{2(r^2+1)}$
    Therefore: .$\displaystyle \boxed{\theta \;=\;\tan^{-1}\!\left(\frac{3r}{2(r^2+1)}\right)}$

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