Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 4"
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If <math>(1 + i)^{100}</math> is expanded and written in the form <math>a + bi</math> where <math>a</math> and <math>b</math> are real numbers, then <math>a =</math> | If <math>(1 + i)^{100}</math> is expanded and written in the form <math>a + bi</math> where <math>a</math> and <math>b</math> are real numbers, then <math>a =</math> | ||
| − | <center><math> \mathrm{(A) \ }-2^{50} \qquad \mathrm{(B) \ }20^{50}-\frac{100!}{50!50!} \qquad \mathrm{(C) \ } \frac{100!}{(25!)^ | + | <center><math> \mathrm{(A) \ } -2^{50} \qquad \mathrm{(B) \ } 20^{50} - \frac{100!}{50!50!} \qquad \mathrm{(C) \ } \frac{100!}{(25!)^2 50!} \qquad \mathrm{(D) \ } 100! \left(-\frac 1{50!50!} + \frac 1{25!75!}\right) \qquad \mathrm{(E) \ } 0 </math></center> |
== Solution == | == Solution == | ||
Revision as of 22:38, 21 July 2006
Problem
If 
is expanded and written in the form 
where 
and 
are real numbers, then 
Solution
Notice that 
. We then have 
.
