Difference between revisions of "1989 AHSME Problems/Problem 29"
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==Problem== | ==Problem== | ||
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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+ | (A) <math>-2^{50}</math> (B) <math>-2^{49}</math> (C) 0 (D) <math>2^{49}</math> (E) <math>2^{50}</math> | ||
==Solution== | ==Solution== | ||
− | {{ | + | 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== | ||
+ | {{AHSME box|year=1989|num-b=28|num-a=30}} | ||
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[[Category:Intermediate Combinatorics Problems]] | [[Category:Intermediate Combinatorics Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 19:52, 30 December 2020
Problem
What is the value of the sum
(A) (B)
(C) 0 (D)
(E)
Solution
By the Binomial Theorem,
.
Using the fact that ,
,
,
, and
, the sum becomes:
.
So, .
Using De Moivre's Theorem, .
And finally, .
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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