The value of `1+i+i^(2)+... + i^(n)` is (i) positive (ii) negative (iii) 0 (iv) cannot be determined

Options

A. positive
B. negative
C. 0
D. cannot be determined

Answer

To solve the problem of finding the value of the expression \(1 + i + i^2 + \ldots + i^n\), we can follow these steps:

### Step 1: Understand the powers of \(i\)
The powers of \(i\) (where \(i = \sqrt{-1}\)) cycle every four terms:
- \(i^0 = 1\)
- \(i^1 = i\)
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\) (and the cycle repeats)

### Step 2: Write the sum explicitly
We can express the sum as:
\[ S = 1 + i + i^2 + i^3 + i^4 + \ldots + i^n \]

### Step 3: Determine the number of complete cycles
Since the powers of \(i\) repeat every four terms, we can determine how many complete cycles of four fit into \(n\):
- Let \(k = \left\lfloor \frac{n}{4} \right\rfloor\) (the number of complete cycles)
- Let \(r = n \mod 4\) (the remainder when \(n\) is divided by 4)

### Step 4: Calculate the sum based on \(r\)
The sum can be broken down into complete cycles and the remaining terms:
\[ S = k \cdot (1 + i + (-1) + (-i)) + \text{(remaining terms)} \]
Since \(1 + i - 1 - i = 0\), the contribution from complete cycles is \(0\).

Now, we need to add the remaining terms based on the value of \(r\):
- If \(r = 0\): \(S = 0\)
- If \(r = 1\): \(S = 1\)
- If \(r = 2\): \(S = 1 + i\)
- If \(r = 3\): \(S = 1 + i - 1 - i = 0\)

### Step 5: Conclusion
The value of \(S\) depends on \(n\):
- If \(n \equiv 0 \mod 4\) or \(n \equiv 3 \mod 4\), then \(S = 0\).
- If \(n \equiv 1 \mod 4\), then \(S = 1\) (positive).
- If \(n \equiv 2 \mod 4\), then \(S = 1 + i\) (which has a real part of 1, hence positive).

Since the value of \(S\) depends on \(n\), we conclude that the value of \(1 + i + i^2 + \ldots + i^n\) cannot be determined without knowing \(n\).

### Final Answer
The correct option is (iv) cannot be determined.