1984 AHSME Problems/Problem 30
Contents
[hide]Problem
For any complex number , is defined to be the real number . If , then equals
Solution 1
Let . Note that
Now we multiply by :
By the geometric series formula, is simply . Therefore
A simple application of De Moivre's Theorem shows that is a ninth root of unity (), so
This shows that . Note that , so .
It's not hard to show that , so the number we seek is equal to .
Now we plug into the fraction:
We multiply the numerator and denominator by and simplify to get
The absolute value of this is
Note that, from double angle formulas, , so . Therefore
Therefore the correct answer is .
Solution 2
Notice that is the primitive root of the equation .
Consider the simple geometric progression,
Differentiating this equation (using the Quotient Rule), we obtain the following equation,
Now, we can evaluate the given expression,
Finally, we can calculate the given modulus,
~ plusone
See Also
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