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i Cycle, the Sum of Powers

\[1+i+i^2+i^3+…+i^{57}\] Do I need to say what the question is?

“Find the sum!” is the immediate question that comes to mind when you start mathematics. When you gain more experience in mathematics, you learn to ask a deeper question:

What is the pattern?

You know, 57 is irrelevant. The only important thing here is the general question of the finding the sum without actually doing all the additions. However, to get to the point of finding the sum without actually doing all the additions, we should experiment with actual additions first! For example, let us find the sum of the first seven terms.   

Please find it before continuing reading. 

Did you notice that to have 7 terms, the last power in the sum should be 6 and not 7, and that means we should calculate \[1+i+i^2+i^3+i^4+i^5+i^6\] and not \[1+i+i^2+i^3+i^4+i^5+i^6+i^7\]. 

This was not a trick. In fact, thinking in terms of the number of terms is the key to the pattern that we are looking for. 

What is the sum of the first two terms? 

What is the sum of the first three terms? 

What is the sum of the first four terms?

What is the sum of the first five terms? The first six? The first seven? 

Let us also include “the sum of the first term”. It is silly, but you see why it is useful. Here are the results starting from “the sum of the first term”, following by “the sum of the first two terms”, then, the sum of the first three term”, and so on. 

\[1,1+i, i, 0, 1, 1+i, i, 0, …\]

See, the sum of the first four terms is 0, the sum of the first eight terms is 0, and so on.  And then, after each 0, we have 1, 1+i, and i. Now if we want to know what is \(1+i+i^2+i^3+…+i^{57}\) we only need to know how many fours goes into 58 (that is the number of terms). This number of fours only matters to us, as long as it helps us to find how many terms remain. In this case, no terms remains. So, the sum is 0. 

\[1+\frac{1}{i}+\frac{1}{i^2}+\frac{1}{i^3}+…+\frac{1}{i^{2019}}\]

Do I need to say what the question is?

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