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Difference between revisions of "1989 AHSME Problems/Problem 29"

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What is the value of the sum <math>S=\sum_{k=0}^{49}(-1)^k\binom{99}{2k}=\binom{99}{0}-\binom{99}{2}+\binom{99}{4}-\cdots -\binom{99}{98}?</math>
 
What is the value of the sum <math>S=\sum_{k=0}^{49}(-1)^k\binom{99}{2k}=\binom{99}{0}-\binom{99}{2}+\binom{99}{4}-\cdots -\binom{99}{98}?</math>
  
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{{incomplete|answers}}
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(A) <math>-2^{50}</math> (B) <math>-2^{49}</math> (C) 0 (D) <math>2^{49}</math> (E) <math>2^{50}</math>
  
 
==Solution==
 
==Solution==
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{{solution}}
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By the [[Binomial Theorem]], <math>(1+i)^{99}=\sum_{n=0}^{99}\binom{99}{j}i^n =</math> <math>\binom{99}{0}i^0+\binom{99}{1}i^1+\binom{99}{2}i^2+\binom{99}{3}i^3+\binom{99}{4}i^4+\cdots +\binom{99}{98}i^{98}</math>.
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Using the fact that <math>i^1=i</math>, <math>i^2=-1</math>, <math>i^3=-i</math>, <math>i^4=1</math>, and <math>i^{n+4}=i^n</math>, the sum becomes:
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<math>(1+i)^{99}=\binom{99}{0}+\binom{99}{1}i-\binom{99}{2}-\binom{99}{3}i+\binom{99}{4}+\cdots -\binom{99}{98}</math>.
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So, <math>Re[(1+i)^{99}]=\binom{99}{0}-\binom{99}{2}+\binom{99}{4}-\cdots -\binom{99}{98} = S</math>.
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Using [[De Moivre's Theorem]], <math>(1+i)^{99}=[\sqrt{2}cis(45^\circ)]^{99}=\sqrt{2^{99}}\cdot cis(99\cdot45^\circ)=2^{49}\sqrt{2}\cdot cis(135^\circ) = -2^{49}+2^{49}i</math>.
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And finally, <math>S=Re[-2^{49}+2^{49}i] = -2^{49}</math>.
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<math>\fbox{B}</math>
  
 
==See also==
 
==See also==
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[[Category:Intermediate Combinatorics Problems]]
 
[[Category:Intermediate Combinatorics Problems]]
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{{MAA Notice}}

Latest revision as of 18:52, 30 December 2020

Problem

What is the value of the sum $S=\sum_{k=0}^{49}(-1)^k\binom{99}{2k}=\binom{99}{0}-\binom{99}{2}+\binom{99}{4}-\cdots -\binom{99}{98}?$


(A) $-2^{50}$ (B) $-2^{49}$ (C) 0 (D) $2^{49}$ (E) $2^{50}$

Solution

By the Binomial Theorem, $(1+i)^{99}=\sum_{n=0}^{99}\binom{99}{j}i^n =$ $\binom{99}{0}i^0+\binom{99}{1}i^1+\binom{99}{2}i^2+\binom{99}{3}i^3+\binom{99}{4}i^4+\cdots +\binom{99}{98}i^{98}$.

Using the fact that $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$, and $i^{n+4}=i^n$, the sum becomes:

$(1+i)^{99}=\binom{99}{0}+\binom{99}{1}i-\binom{99}{2}-\binom{99}{3}i+\binom{99}{4}+\cdots -\binom{99}{98}$.

So, $Re[(1+i)^{99}]=\binom{99}{0}-\binom{99}{2}+\binom{99}{4}-\cdots -\binom{99}{98} = S$.

Using De Moivre's Theorem, $(1+i)^{99}=[\sqrt{2}cis(45^\circ)]^{99}=\sqrt{2^{99}}\cdot cis(99\cdot45^\circ)=2^{49}\sqrt{2}\cdot cis(135^\circ) = -2^{49}+2^{49}i$.

And finally, $S=Re[-2^{49}+2^{49}i] = -2^{49}$.

$\fbox{B}$

See also

1989 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 28 		Followed by
Problem 30
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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