The imaginary number i is defined such that i2 = –1. What does i + i2 + i3 + ... + i23 equal?
did you mean
i + i^2 + ... + i^23 ?
notice this pattern
i^1 = i
i^2 = -1
i^3 = -i
i^4 = +1
i^5 = i
....
notice that
i + i^2 + i^3 + i^4 = 0 and
i^5 + i^6 + i^7 + i^8 = 0
etc
let me know what you got.
Well, that's a whole lot of imaginary numbers! Let's see if we can make sense of it.
We know that i² equals -1, so let's work with that.
i + i² can be written as i + (-1), which simplifies to i - 1.
Now, let's consider what happens when we multiply i - 1 by i. We get (i - 1) * i = i² - i = -1 - i.
If we repeat this process by multiplying (-1 - i) by i, we get (-1 - i) * i = -i - i² = -i - (-1) = -i + 1.
By continuing this pattern, we'll eventually reach i²³.
Now, let's examine what happens as we continue expanding this series:
(i - 1) --> (-1 - i) --> (-i + 1) --> (1 + i) --> ...
We can see that every four terms, the pattern repeats. Therefore, we can break this down into four groups:
Group 1: i - 1
Group 2: -1 - i
Group 3: -i + 1
Group 4: 1 + i
Since we have 23 terms, we can divide 23 by 4, which gives us a remainder of 3, indicating that we are in Group 3.
So, i + i² + i³ + ... + i²³ equals -i + 1.
Now, getting to the punchline, we can finally say:
The sum i + i² + i³ + ... + i²³ equals... wait for it... -i + 1! Now wasn't all that imaginary fun?
To find the value of i + i^2 + i^3 + ... + i^23, we can notice that the powers of i follow a cyclic pattern. Since i^2 = -1, i^3 = i^2 * i = -1 * i = -i, and i^4 = i^2 * i^2 = (-1) * (-1) = 1. Therefore, the powers of i repeat every four terms: i, i^2, i^3, i^4 = i, -1, -i, 1.
Now, we need to determine the remainder when 23 is divided by 4. 23 ÷ 4 = 5 remainder 3. So, the expression can be written as i^(4*5 + 3) = i^3.
Therefore, i + i^2 + i^3 + ... + i^23 = -i + i^2 + i^3 = -i + (-1) + (-i) = -1 - 2i.
To find the value of i + i^2 + i^3 + ... + i^23, we can start by noticing a pattern in the powers of i. Let's calculate the first few powers of i:
i^1 = i
i^2 = i × i = -1
i^3 = i × i × i = -i
i^4 = i × i × i × i = 1
i^5 = i × i × i × i × i = i
i^6 = i × i × i × i × i × i = -1
As we can see, the powers of i repeat in a cycle of four: {i, -1, -i, 1, i, -1, -i, 1, ...}.
Since we want to calculate the sum up to i^23, we need to find out which element in the cycle i^23 corresponds to. We can divide 23 by 4 and observe the remainder:
23 ÷ 4 = 5 remainder 3
This means that i^23 is located at the third position in the cycle, which is -i. Thus, we can rewrite the expression as:
i + i^2 + i^3 + ... + i^23 = i + (-1) + (-i) = -1
Therefore, the sum of i + i^2 + i^3 + ... + i^23 equals -1.