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Difference between revisions of "2019 AMC 8 Problems/Problem 17"

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==Problem 17==
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==Problem==
 
What is the value of the product  
 
What is the value of the product  
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<cmath>\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?</cmath>
 
<cmath>\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?</cmath>
  
 
<math>\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50</math>
 
<math>\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50</math>
  
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==Solution 1==
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==Solution 1 (telescoping)==
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As you can easily see, we can write the product as <cmath>\frac{1}{2}\left(\frac{3\cdot2}{2\cdot3}\right)\left(\frac{4\cdot3}{3\cdot4}\right)\cdots\left(\frac{99\cdot98}{98\cdot99}\right)\cdot\frac{99}{100}.</cmath>
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We rewrite: <cmath>\frac{1}{2}\cdot\left(\frac{3\cdot2}{2\cdot3}\right)\left(\frac{4\cdot3}{3\cdot4}\right)\cdots\left(\frac{99\cdot98}{98\cdot99}\right)\cdot\frac{100}{99}</cmath>
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 +
The middle terms cancel, leaving us with
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 +
<cmath>\left(\frac{1\cdot100}{2\cdot99}\right)= \boxed{\textbf{(B)}\frac{50}{99}}</cmath>
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 +
==Solution 2==
 +
If you calculate the first few values of the equation, all of the values tend to close to <math>\frac{1}{2}</math>, but are not equal to it. The answer closest to <math>\frac{1}{2}</math> but not equal to it is  <math>\boxed{\textbf{(B)}\frac{50}{99}}</math>.
 +
 
 +
==Solution 3==
 +
Rewriting the numerator and the denominator, we get <math>\frac{\frac{100! \cdot 98!}{2}}{\left(99!\right)^2}</math>. We can simplify by canceling 99! on both sides, leaving us with: <math>\frac{100 \cdot 98!}{2 \cdot 99!}</math> We rewrite <math>99!</math> as <math>99 \cdot 98!</math> and cancel <math>98!</math>, which gets <math>\boxed{\textbf{(B)}\frac{50}{99}}</math>.
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 +
==Solution 4==
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All of the terms have the form <math>\frac{k^2-1}{k^2}</math>, which is <math><1</math>, so the product is <math><1</math>, so we eliminate options (D) and (E). (C) is too close to 1 to be possible. The partial products seem to be approaching 1/2, so we guess that 1/2 is the limit/asymptote, and so any finite product would be slightly larger than 1/2. Therefore, by process of elimination and a small guess, we get that the answer is <math>\boxed{\textbf{(B)}\frac{50}{99}}</math>.
 +
 
 +
==Video Solution ==
 +
 
 +
==Video Solution by Math-X (First fully understand the problem!!!)==
 +
https://youtu.be/IgpayYB48C4?si=UJVe2zopeqT-4rLM&t=5256
 +
 
 +
~Math-X
 +
 
 +
https://www.youtube.com/watch?v=yPQmvyVyvaM
 +
 
 +
Associated video
 +
 
 +
https://www.youtube.com/watch?v=ffHl1dAjs7g&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=1
 +
 
 +
~ MathEx
 +
 
 +
== Video Solution 2==
 +
 
 +
Solution detailing how to solve the problem:
 +
 
 +
https://www.youtube.com/watch?v=VezsRMJvGPs&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=18
 +
 
 +
==Video Solution 3==
 +
https://youtu.be/e1EJNZu-jxM
 +
 
 +
~savannahsolver
 +
 
 +
==Video Solution 3(an Elegant way)==
 +
https://www.youtube.com/watch?v=la3en2tgBN0
 +
 
 +
==Video Solution 4 by OmegaLearn==
 +
https://youtu.be/TkZvMa30Juo?t=3326
 +
 
 +
~ pi_is_3.14
 +
 
 +
==Video Solution ==
 +
https://youtu.be/wUvi7tzxuTk
  
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If we remove all parentheses, we can easily see that the middle terms cancel, leaving us with
+
~Education, the Study of Everything
  
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<cmath>\left(\frac{1\cdot100}{2\cdot99}\right)</cmath> = <math>\boxed{\textbf{(B)}\frac{50}{99}}</math>
+
==Video Solution by The Power of Logic(1 to 25 Full Solution)==
 +
https://youtu.be/Xm4ZGND9WoY
  
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~phoenixfire & dreamr
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~Hayabusa1
  
 
==See Also==
 
==See Also==

Latest revision as of 09:32, 9 November 2024

Problem

What is the value of the product

\[\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?\]

$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50$

Solution 1 (telescoping)

We rewrite: \[\frac{1}{2}\cdot\left(\frac{3\cdot2}{2\cdot3}\right)\left(\frac{4\cdot3}{3\cdot4}\right)\cdots\left(\frac{99\cdot98}{98\cdot99}\right)\cdot\frac{100}{99}\]

The middle terms cancel, leaving us with

\[\left(\frac{1\cdot100}{2\cdot99}\right)= \boxed{\textbf{(B)}\frac{50}{99}}\]

Solution 2

If you calculate the first few values of the equation, all of the values tend to close to $\frac{1}{2}$, but are not equal to it. The answer closest to $\frac{1}{2}$ but not equal to it is $\boxed{\textbf{(B)}\frac{50}{99}}$.

Solution 3

Rewriting the numerator and the denominator, we get $\frac{\frac{100! \cdot 98!}{2}}{\left(99!\right)^2}$. We can simplify by canceling 99! on both sides, leaving us with: $\frac{100 \cdot 98!}{2 \cdot 99!}$ We rewrite $99!$ as $99 \cdot 98!$ and cancel $98!$, which gets $\boxed{\textbf{(B)}\frac{50}{99}}$.

Solution 4

All of the terms have the form $\frac{k^2-1}{k^2}$, which is $<1$, so the product is $<1$, so we eliminate options (D) and (E). (C) is too close to 1 to be possible. The partial products seem to be approaching 1/2, so we guess that 1/2 is the limit/asymptote, and so any finite product would be slightly larger than 1/2. Therefore, by process of elimination and a small guess, we get that the answer is $\boxed{\textbf{(B)}\frac{50}{99}}$.

Video Solution

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/IgpayYB48C4?si=UJVe2zopeqT-4rLM&t=5256

~Math-X

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

Associated video

https://www.youtube.com/watch?v=ffHl1dAjs7g&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=1

~ MathEx

Video Solution 2

Solution detailing how to solve the problem:

https://www.youtube.com/watch?v=VezsRMJvGPs&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=18

Video Solution 3

https://youtu.be/e1EJNZu-jxM

~savannahsolver

Video Solution 3(an Elegant way)

https://www.youtube.com/watch?v=la3en2tgBN0

Video Solution 4 by OmegaLearn

https://youtu.be/TkZvMa30Juo?t=3326

~ pi_is_3.14

Video Solution

https://youtu.be/wUvi7tzxuTk

~Education, the Study of Everything

Video Solution by The Power of Logic(1 to 25 Full Solution)

https://youtu.be/Xm4ZGND9WoY

~Hayabusa1

See Also

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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