Difference between revisions of "Imaginary unit/Introductory"
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| − | #Find the sum of <math>i^1+i^2+\ldots+i^{2006}</math>. | + | #Find the sum of <math>i^1+i^2+\ldots+i^{2006}</math>. |
| − | #Find the product of <math>i^1 \times i^2 \times \cdots \times i^{2006}</math>. | + | #Find the product of <math>i^1 \times i^2 \times \cdots \times i^{2006}</math>. |
__TOC__ | __TOC__ | ||
Revision as of 08:52, 27 October 2007
- Find the sum of
. - Find the product of
.
.
.Contents
Solution 1
Since 
repeats in a n exponential series at every fifth turn, the sequence i, -1, -i, 1 repeats. Note that this sums to 0. That means that all sequences 
have a sum of zero (k is a natural number). Since 
, the original series sums to the first two terms of the powers of i, which equals 
.
Solution 2
, so the product is equal to 
.
