SOLUTION: The imaginary number i is defined such that i^2 = –1. What does i + i^2 + i^3 + .... + i^23 equal? The answer is supposed to be -1, but I thought it should be -i according to anoth

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: The imaginary number i is defined such that i^2 = –1. What does i + i^2 + i^3 + .... + i^23 equal? The answer is supposed to be -1, but I thought it should be -i according to anoth      Log On


   

Question 176697: The imaginary number i is defined such that i^2 = –1. What does i + i^2 + i^3 + .... + i^23 equal? The answer is supposed to be -1, but I thought it should be -i according to another source. Is the answer key wrong?
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
i%5E1+=+i

i%5E2+=+-1

i%5E3+=+i%5E2+%2A+i+=+-1+%2A+i+=+-i

i%5E4+=+i%5E2+%2A+i%5E2+=+-1+%2A+-1+=+1

i%5E5+=+i%5E4+%2A+i+=+1+%2A+i+=+i, and so on repeating the sequence every 4 increments of the exponent.

The sum of the first 4 is: i+%2B+%28-1%29+%2B+%28-i%29+%2B+1+=+0, so the sum of the each subsequent 4 must also be zero.

Use integer division 23%2F4 is 5 with a remainder of 3. So in your string of 23 terms there are 5 sets of 4 terms each of which sum to zero, so it is only the last three terms that make any difference:

i+%2B+%28-1%29+%2B+%28-i%29+=+-1

And that is your sum. It is your 'other source' that is incorrect.