Difference between revisions of "1989 AHSME Problems/Problem 29"
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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>. | 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>. | ||
− | And finally, <math>S=Re[-2^{49}+2^{49}i] = -2^{49}</math>. | + | And finally, <math>S=Re[-2^{49}+2^{49}i] = -2^{49}</math>. |
+ | |||
+ | <math>\fbox{B}</math> | ||
==See also== | ==See also== |
Revision as of 14:25, 7 October 2014
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.