Difference between revisions of "1992 AHSME Problems/Problem 15"
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== Problem == | == Problem == | ||
| − | Let <math> | + | Let <math>i=\sqrt{-1}</math>. Define a sequence of complex numbers by |
<cmath>z_1=0,\quad z_{n+1}=z_{n}^2+i \text{ for } n\ge1.</cmath> | <cmath>z_1=0,\quad z_{n+1}=z_{n}^2+i \text{ for } n\ge1.</cmath> | ||
| Line 13: | Line 13: | ||
== Solution == | == Solution == | ||
| − | <math>\fbox{ | + | <math>\fbox{B}</math> Write out some terms: <math>0, i, -1+i, -i, -1+i, -i, -1+i, -i</math>, etc., and it keeps alternating between <math>-1+i</math> and <math>-i</math>, so as <math>111</math> is odd, <math>z_{111}</math> is <math>-1+i</math>. Thus its distance from the origin is <math>\sqrt{(-1)^2+1^2} = \sqrt{2}</math>. |
== See also == | == See also == | ||
Latest revision as of 01:46, 20 February 2018
Problem
Let 
. Define a sequence of complex numbers by
![\[z_1=0,\quad z_{n+1}=z_{n}^2+i \text{ for } n\ge1.\]](http://latex.artofproblemsolving.com/4/5/3/453863d57253cb6e8dabf7f2f486b89f70402788.png)
In the complex plane, how far from the origin is 
?
Solution
Write out some terms: 
, etc., and it keeps alternating between 
and 
, so as 
is odd, is . Thus its distance from the origin is 
.
See also
| 1992 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
| 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. 
