Difference between revisions of "2008 AMC 12A Problems/Problem 25"

Latest revision as of 22:13, 2 September 2024

Problem

A sequence $(a_1,b_1)$ , $(a_2,b_2)$ , $(a_3,b_3)$ , $\ldots$ of points in the coordinate plane satisfies

$(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)$ for $n = 1,2,3,\ldots$ .

Suppose that $(a_{100},b_{100}) = (2,4)$ . What is $a_1 + b_1$ ?

$\mathrm{(A)}\ -\frac{1}{2^{97}}\qquad\mathrm{(B)}\ -\frac{1}{2^{99}}\qquad\mathrm{(C)}\ 0\qquad\mathrm{(D)}\ \frac{1}{2^{98}}\qquad\mathrm{(E)}\ \frac{1}{2^{96}}$

Solution 1

This sequence can also be expressed using matrix multiplication as follows:

$\left[ \begin{array}{c} a_{n+1} \\ b_{n+1} \end{array} \right] = \left[ \begin{array}{cc} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{array} \right] \left[ \begin{array}{c} a_{n} \\ b_{n} \end{array} \right] = 2 \left[ \begin{array}{cc} \cos 30^\circ & -\sin 30^\circ \\ \sin 30^\circ & \ \cos 30^\circ \end{array} \right] \left[ \begin{array}{c} a_{n} \\ b_{n} \end{array} \right]$ .

Thus, $(a_{n+1} , b_{n+1})$ is formed by rotating $(a_n , b_n)$ counter-clockwise about the origin by $30^\circ$ and dilating the point's position with respect to the origin by a factor of $2$ .

So, starting with $(a_{100},b_{100})$ and performing the above operations $99$ times in reverse yields $(a_1,b_1)$ .

Rotating $(2,4)$ clockwise by $99 \cdot 30^\circ \equiv 90^\circ$ yields $(4,-2)$ . A dilation by a factor of $\frac{1}{2^{99}}$ yields the point $(a_1,b_1) = \left(\frac{4}{2^{99}}, -\frac{2}{2^{99}} \right) = \left(\frac{1}{2^{97}}, -\frac{1}{2^{98}} \right)$ .

Therefore, $a_1 + b_1 = \frac{1}{2^{97}} - \frac{1}{2^{98}} = \frac{1}{2^{98}} \Rightarrow D$ .

Solution 2 (algebra)

Let $(x,y)=(a_1,b_1)$ . Then, we can begin to list out terms as follows:

$(a_2,b_2)=(x\sqrt{3}-y,y\sqrt{3}+x)$

$(a_3,b_3)=(2x-2y\sqrt{3},2y+2x\sqrt{3})$

$(a_4,b_4)=(-8y,8x)$

We notice that the sequence follows the rule $(a_{n+3},b_{n+3})=(-2^3b_n,2^3a_n)$

We can now start listing out every third point, getting:

$(a_1,b_1)=(x,y)$

$(a_4,b_4)=(-2^3y,2^3x)$

$(a_7,b_7)=(-2^6x,-2^6y)$

$(a_{10},b_{10})=(2^9y,-2^9x)$

$(a_{13},b_{13})=(2^{12}x,2^{12}y)$

We can make two observations from this:

(1) In $a_n$ , the coefficient of $x$ and $y$ is $2^{n-1}$

(2) The positioning of $x$ and $y$ , and their signs, cycle with every $12$ terms.

We know then that from (1), the coefficients of $x$ and $y$ in $(a_{100},b_{100})$ are both $2^{99}$

We can apply (2), finding $100 \text{(mod }12)=4$ , so the positions and signs of $x$ and $y$ are the same in $(a_{100},b_{100})$ as they are in $(a_{4},b_{4})$ .

From this, we can get $(a_{100},b_{100})=(-2^{99}y,2^{99}x)$ . We know that $(a_{100},b_{100})=(2,4)$ , so we get the following:

$-2^{99}y=2 \Rightarrow y=-\frac{1}{2^{98}}$

$2^{99}x=4 \Rightarrow x=\frac{1}{2^{97}}$

The answer is $x+y=\frac{1}{2^{97}}-\frac{1}{2^{98}}=\boxed{\textbf{(D) }\frac{1}{2^{98}}}$ ..

Solution 3

The ordered pairs and $\sqrt{3}$ 's makes us think to use complex numbers. We have $(a_{n+1},b_{n+1}) = 2\left(\frac{\sqrt{3}}{2}a_n - \frac{1}{2}b_n, \frac{\sqrt{3}}{2}b_n + \frac{1}{2}a_n\right)$ , so $a_{n+1} + b_{n+1}i = 2\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right)(a_n + b_ni) = \frac{1}{2}e^{i\pi/6}(a_n + b_ni)$ . Letting $z_n = a_n + b_ni$ (so $z_{n+1} = a_{n+1} + b_{n+1}i$ ), we have $z_{n+1} = 2e^{i\pi/6}z_n$ . Letting $n\rightarrow n-1$ , we have $z_n = 2e^{i\pi/6}z_{n-1}$ , so $z_{n-1} = \frac{1}{2}e^{-i\pi/6}z_n$ . This is the reverse transformation. We have $z_{99} = \frac{1}{2}e^{-i\pi/6}z_{100}$ $z_{98} = \frac{1}{2^2}e^{2(-i\pi/6)}z_{100}$ $\vdots$ $z_{1} = \frac{1}{2^{99}}e^{99(-i\pi/6)}z_{100}$ $= \frac{1}{2^{99}}e^{-i\pi/2}z_{100} = -\frac{1}{2^{99}}i(2 + 4i) = \frac{1}{2^{97}} - \frac{1}{2^{98}}i.$

Hence, $a_1 + b_1 = \frac{1}{2^{97}} - \frac{1}{2^{98}} = \boxed{\mathbf{(D)}\frac{1}{2^{98}}}$ ~ brainfertilzer.

Solution 4 (Kinda braindead)

Start by turning the two equations into $a_n = \frac{a_{n+1}\sqrt{3}+b_{n+1}}{4}$ and $b_n = \frac{b_{n+1}\sqrt{3}-a_{n+1}}{4}$ . Note that these are just obtained by solving the equations.

This makes finding values of $a_n$ and $b_n$ much easier, and soon we notice that $a_{97} = \frac{1}{2}$ and $b_{97} = -\frac{1}{4}$ . After that, we get that $a_{94} = -\frac{1}{32}$ and $b_{94} = -\frac{1}{16}$ . Observe that $|a_n| = |\frac{b_{n+3}}{8}|$ and $|b_n| = |\frac{a_{n+3}}{8}|$ . This is basically just ignoring signs.

Now, we proceed to find $|a_1| = \frac{1}{2^{97}}$ while $|b_1| = \frac{1}{2^{98}}$ . Despite there being 4 possible sign combinations for $(a_1, b_1)$ , the only achievable answer choice is $\boxed{\textbf{(D)}\frac{1}{2^{98}}}$

-skibbysiggy

Video Solution

https://www.youtube.com/watch?v=_4UJzyBslFA

@@ Line 8: / Line 8: @@
 <math>\mathrm{(A)}\ -\frac{1}{2^{97}}\qquad\mathrm{(B)}\ -\frac{1}{2^{99}}\qquad\mathrm{(C)}\ 0\qquad\mathrm{(D)}\ \frac{1}{2^{98}}\qquad\mathrm{(E)}\ \frac{1}{2^{96}}</math>
-==Solution==
+==Solution 1==
 This sequence can also be expressed using matrix multiplication as follows:
@@ Line 21: / Line 21: @@
 Therefore, <math>a_1 + b_1 = \frac{1}{2^{97}} - \frac{1}{2^{98}} = \frac{1}{2^{98}} \Rightarrow D</math>.
-Shortcut: no answer has <math>3</math> in the denominator. So the point cannot have orientation <math>(2,4)</math> or <math>(-2,-4)</math>. Also there are no negative answers. Any other non-multiple of <math>90^\circ</math> rotation of <math>30n^\circ</math> would result in the need of radicals. So either it has orientation <math>(4,-2)</math> or <math>(-4,2)</math>. Both answers add up to <math>2</math>. Thus, <math>2/2^99=\boxed{\textbf{(D) }\frac{1}{2^{98}}}</math>.
+==Solution 2 (algebra)==
+Let <math>(x,y)=(a_1,b_1)</math>. Then, we can begin to list out terms as follows:
+<math>(a_2,b_2)=(x\sqrt{3}-y,y\sqrt{3}+x)</math>
+<math>(a_3,b_3)=(2x-2y\sqrt{3},2y+2x\sqrt{3})</math>
+<math>(a_4,b_4)=(-8y,8x)</math>
+We notice that the sequence follows the rule <math>(a_{n+3},b_{n+3})=(-2^3b_n,2^3a_n)</math>
+We can now start listing out every third point, getting:
+<math>(a_1,b_1)=(x,y)</math>
+<math>(a_4,b_4)=(-2^3y,2^3x)</math>
+<math>(a_7,b_7)=(-2^6x,-2^6y)</math>
+<math>(a_{10},b_{10})=(2^9y,-2^9x)</math>
+<math>(a_{13},b_{13})=(2^{12}x,2^{12}y)</math>
+We can make two observations from this:
+(1) In <math>a_n</math>, the coefficient of <math>x</math> and <math>y</math> is <math>2^{n-1}</math>
+(2) The positioning of <math>x</math> and <math>y</math>, and their signs, cycle with every <math>12</math> terms.
+We know then that from (1), the coefficients of <math>x</math> and <math>y</math> in <math>(a_{100},b_{100})</math> are both <math>2^{99}</math>
+We can apply (2), finding <math>100 \text{(mod }12)=4</math>, so the positions and signs of <math>x</math> and <math>y</math> are the same in <math>(a_{100},b_{100})</math> as they are in <math>(a_{4},b_{4})</math>.
+From this, we can get <math>(a_{100},b_{100})=(-2^{99}y,2^{99}x)</math>. We know that <math>(a_{100},b_{100})=(2,4)</math>, so we get the following:
+<math>-2^{99}y=2 \Rightarrow y=-\frac{1}{2^{98}}</math>
+<math>2^{99}x=4 \Rightarrow x=\frac{1}{2^{97}}</math>
+The answer is <math>x+y=\frac{1}{2^{97}}-\frac{1}{2^{98}}=\boxed{\textbf{(D) }\frac{1}{2^{98}}}</math>..
+==Solution 3==
+The ordered pairs and <math>\sqrt{3}</math>'s makes us think to use complex numbers. We have <math>(a_{n+1},b_{n+1}) = 2\left(\frac{\sqrt{3}}{2}a_n - \frac{1}{2}b_n, \frac{\sqrt{3}}{2}b_n + \frac{1}{2}a_n\right)</math>, so <math>a_{n+1} + b_{n+1}i = 2\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right)(a_n + b_ni) = \frac{1}{2}e^{i\pi/6}(a_n + b_ni)</math>. Letting <math>z_n = a_n + b_ni</math> (so <math>z_{n+1} = a_{n+1} + b_{n+1}i</math>), we have <math>z_{n+1} = 2e^{i\pi/6}z_n</math>. Letting <math>n\rightarrow n-1</math>, we have <math>z_n = 2e^{i\pi/6}z_{n-1}</math>, so <math>z_{n-1} = \frac{1}{2}e^{-i\pi/6}z_n</math>. This is the reverse transformation. We have
+<cmath> z_{99} = \frac{1}{2}e^{-i\pi/6}z_{100}</cmath>
+<cmath> z_{98} = \frac{1}{2^2}e^{2(-i\pi/6)}z_{100}</cmath>
+<cmath> \vdots</cmath>
+<cmath> z_{1} = \frac{1}{2^{99}}e^{99(-i\pi/6)}z_{100}</cmath>
+<cmath> = \frac{1}{2^{99}}e^{-i\pi/2}z_{100} = -\frac{1}{2^{99}}i(2 + 4i) = \frac{1}{2^{97}} - \frac{1}{2^{98}}i.</cmath>
+Hence, <math>a_1 + b_1 = \frac{1}{2^{97}} - \frac{1}{2^{98}} = \boxed{\mathbf{(D)}\frac{1}{2^{98}}}</math> ~ brainfertilzer.
+==Solution 4 (Kinda braindead)==
+Start by turning the two equations into <math>a_n = \frac{a_{n+1}\sqrt{3}+b_{n+1}}{4}</math> and <math>b_n = \frac{b_{n+1}\sqrt{3}-a_{n+1}}{4}</math>. Note that these are just obtained by solving the equations.
+This makes finding values of <math>a_n</math> and <math>b_n</math> much easier, and soon we notice that <math>a_{97} = \frac{1}{2}</math> and <math>b_{97} = -\frac{1}{4}</math>. After that, we get that <math>a_{94} = -\frac{1}{32}</math> and <math>b_{94} = -\frac{1}{16}</math>. Observe that <math>|a_n| = |\frac{b_{n+3}}{8}|</math> and <math>|b_n| = |\frac{a_{n+3}}{8}|</math>. This is basically just ignoring signs.
+Now, we proceed to find <math>|a_1| = \frac{1}{2^{97}}</math> while <math>|b_1| = \frac{1}{2^{98}}</math>. Despite there being 4 possible sign combinations for <math>(a_1, b_1)</math>, the only achievable answer choice is <math>\boxed{\textbf{(D)}\frac{1}{2^{98}}}</math>
+-skibbysiggy
+==Video Solution==
+https://www.youtube.com/watch?v=_4UJzyBslFA
 ==See Also==
 {{AMC12 box|year=2008|ab=A|num-b=24|after=Last question}}
 {{MAA Notice}}

2008 AMC 12A (Problems • Answer Key • Resources)
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