Simplify
i^1+i^2+i^3+... i^97 + i^98+i^99
Answer= -1
i-1-i+1+i-1 ......
To simplify the given expression, we need to determine the pattern of powers of "i" and find a way to combine them.
The imaginary unit "i" is defined as the square root of -1. It has the following pattern of powers:
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
As we can see, the powers of "i" repeat every four terms: i, -1, -i, 1.
Now, let's group the given terms in the expression based on this pattern:
(i^1 + i^5 + i^9 + ... + i^97) + (i^2 + i^6 + i^10 + ... + i^98) + (i^3 + i^7 + i^11 + ... + i^99) + i^4
In each group, we can factor out "i" since the pattern starts with "i":
i * (1 + i^4 + i^8 + ... + i^96) + i^2 * (1 + i^4 + i^8 + ... + i^96) + i^3 * (1 + i^4 + i^8 + ... + i^96) + i^4
Now, let's simplify each group:
i * [1 + i^4 * (1 + i^4 + i^8 + ... + i^92)] + i^2 * [1 + i^4 * (1 + i^4 + i^8 + ... + i^92)] + i^3 * [1 + i^4 * (1 + i^4 + i^8 + ... + i^92)] + i^4
Since i^4 is equal to 1, we can further simplify the expression:
i * (1 + 1 + 1 + ... + 1) + i^2 * (1 + 1 + 1 + ... + 1) + i^3 * (1 + 1 + 1 + ... + 1) + 1
Now, let's count the number of terms in each group. In each group, there are 24 terms (i^4 is equal to 1, so the sum is just 24). Therefore, we have:
i * 24 + i^2 * 24 + i^3 * 24 + 1
Simplifying further:
24i + 24(-1) + 24(-i) + 1
Combining like terms:
(24i - 24i) + (24(-1) + 1)
The imaginary terms cancel out, and we're left with:
-24 + 1
Finally:
-23
So the simplified form of the expression i^1 + i^2 + i^3 + ... + i^97 + i^98 + i^99 is -23.