Find the sum (`i + i^(2) + i^(3) + i^(4) +.....` up to 400 terms).

Explanation

To find the sum \( S = i + i^2 + i^3 + i^4 + \ldots \) up to 400 terms, we first need to understand the powers of \( i \).

1. **Identify the powers of \( i \)**:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

After \( i^4 \), the powers of \( i \) repeat every 4 terms:
- \( i^5 = i \)
- \( i^6 = -1 \)
- \( i^7 = -i \)
- \( i^8 = 1 \)
- and so on...

2. **Group the terms**:
Since the powers of \( i \) repeat every 4 terms, we can group the terms in sets of 4:
\[ (i + i^2 + i^3 + i^4) + (i + i^2 + i^3 + i^4) + \ldots \]
Each group of 4 terms sums to:
\[ i + (-1) + (-i) + 1 = 0 \]

3. **Determine how many complete groups of 4 are in 400 terms**:
The total number of terms is 400. Since each group contains 4 terms, the number of complete groups is:
\[ \frac{400}{4} = 100 \]

4. **Calculate the total sum**:
Since each group sums to 0, the total sum \( S \) is:
\[ S = 100 \times 0 = 0 \]

Thus, the sum \( i + i^2 + i^3 + i^4 + \ldots \) up to 400 terms is \( \boxed{0} \).