Write the value of `i+i^(10)+i^(20)+i^(30)`

Answer

To solve the expression \( i + i^{10} + i^{20} + i^{30} \), we first need to understand the powers of \( i \), where \( i \) is defined as \( \sqrt{-1} \).

### Step-by-step Solution:

1. **Identify the powers of \( i \)**:
The powers of \( i \) cycle every four terms:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \) (and the cycle repeats)

2. **Calculate \( i^{10} \)**:
To find \( i^{10} \), we can use the cycle:
\[ 10 \mod 4 = 2 \quad \text{(since 10 divided by 4 gives a remainder of 2)} \]
Therefore, \( i^{10} = i^2 = -1 \).

3. **Calculate \( i^{20} \)**:
Similarly, for \( i^{20} \):
\[ 20 \mod 4 = 0 \quad \text{(since 20 divided by 4 gives a remainder of 0)} \]
Thus, \( i^{20} = i^0 = 1 \).

4. **Calculate \( i^{30} \)**:
For \( i^{30} \):
\[ 30 \mod 4 = 2 \quad \text{(since 30 divided by 4 gives a remainder of 2)} \]
So, \( i^{30} = i^2 = -1 \).

5. **Substitute the values back into the expression**:
Now we can substitute the calculated values back into the expression:
\[ i + i^{10} + i^{20} + i^{30} = i + (-1) + 1 + (-1) \]

6. **Simplify the expression**:
Simplifying the above expression:
\[ i - 1 + 1 - 1 = i - 1 \]

Thus, the final value of \( i + i^{10} + i^{20} + i^{30} \) is:
\[ \boxed{i - 1} \]