Author |
Topic: x to the 2005 (Read 1346 times) |
|
ThudnBlunder
Uberpuzzler
The dewdrop slides into the shining Sea
Gender: 
Posts: 4489
|
 |
x to the 2005
« on: Feb 1st, 2005, 9:38am » |
 Quote  Modify
|
If x + 1/x = -1 find the value of x2005 + 1/x2005
|
|
 IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
Barukh
Uberpuzzler
Gender:
Posts: 2276
|
|
Re: x to the 2005
« Reply #1 on: Feb 1st, 2005, 10:11am » |
Quote Modify
|
[smiley=blacksquare.gif]
Rewriting the equation in the form x2 + x + 1 = 0, we see that x is one of the 2 complex cubic roots of unity, while 1/x is the other. WLOG, let x = e2[pi]i/3, then 1/x = e4[pi]i/3.
Referring to the plane of complex numbers, we see that raising x to the n-th power is a rotation of the point (1,0) counter-clockwise by the angle 2[pi]n/3, and analogously for 1/x. When n [ne] 0 mod 3, xn + 1/xn = x + 1/x = -1. When n = 0 mod 3, xn = 1/xn = 1.
Since 3 doesn’t divide 2005, the answer is -1.
[smiley=blacksquare.gif]
|
| « Last Edit: Feb 1st, 2005, 10:13am by Barukh » |
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print