Find the value of 1+i^{2}+i^{4}+i^{6}+i^{8}+...+i^{20}


Please select simplest and short method to do it 

The given expression involves powers of \( i \) (the imaginary unit). The key to solving this is knowing the cyclical pattern of powers of \( i \):

\[
i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
\]

These powers repeat every four terms, forming a cycle: \( i, -1, -i, 1 \).

Now, let's break down the given sum:
\[
1 + i^2 + i^4 + i^6 + i^8 + \dots + i^{20}
\]

### Step-by-step approach:

1. First, note that \( i^2 = -1 \) and every fourth power of \( i \) is 1. So the sum is composed of the terms \( 1, i^2, i^4, i^6, \dots, i^{20} \).

2. The powers of \( i \) repeat in cycles of 4. So we can group the terms into sets of 4:
   \[
   (1 + i^2 + i^4 + i^6) + (i^8 + i^{10} + i^{12} + i^{14}) + (i^{16} + i^{18} + i^{20})
   \]
   
3. Using the cyclical pattern:
   - \( 1 + i^2 + i^4 + i^6 = 1 + (-1) + 1 + (-1) = 0 \)
   - Similarly, \( i^8 + i^{10} + i^{12} + i^{14} = 1 + (-1) + 1 + (-1) = 0 \)
   - \( i^{16} + i^{18} + i^{20} = 1 + (-1) + 1 = 1 \)

4. Therefore, the total sum is:
   \[
   0 + 0 + 1 = 1
   \]

### Final answer:
\[
\boxed{1}
\]

Let me know if you'd like a more detailed explanation or have further questions!

**Related Questions:**
1. What is the value of \( 1 + i^1 + i^2 + i^3 + i^4 + \dots + i^{20} \)?
2. How does the periodicity of powers of \( i \) help in simplifying large sums?
3. What is the sum of \( i^n \) for odd powers \( n \) up to 20?
4. How can you generalize the sum of powers of \( i \) up to any \( n \)?
5. Why does \( i^4 = 1 \) and how does this help in cyclic calculations?

**Tip:** When dealing with powers of imaginary numbers, look for repeating cycles to simplify the calculations.

Find the value of 1+i^{2}+i^{4}+i^{6}+i^{8}+...+i^{20}.

Learn how to solve a cyclic power series involving the imaginary unit i, such as 1+i^2+i^4+...+i^20. This problem demonstrates the cyclical nature of powers of i and provides a step-by-step solution using key concepts in complex numbers. Suitable for students in Grades 10-12.

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