Find the value of `i^(248) + i^(341) + i^(442) + i^(543)`.

Answer

To solve the expression \( i^{248} + i^{341} + i^{442} + i^{543} \), we first need to understand the powers of \( i \), where \( i = \sqrt{-1} \).

### Step 1: Determine the pattern of 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)

### Step 2: Find \( i^{248} \)
To find \( i^{248} \), we calculate \( 248 \mod 4 \):
\[ 248 \div 4 = 62 \quad \text{(remainder 0)} \]
Thus, \( i^{248} = i^0 = 1 \).

### Step 3: Find \( i^{341} \)
Next, we calculate \( 341 \mod 4 \):
\[ 341 \div 4 = 85 \quad \text{(remainder 1)} \]
Thus, \( i^{341} = i^1 = i \).

### Step 4: Find \( i^{442} \)
Now, we calculate \( 442 \mod 4 \):
\[ 442 \div 4 = 110 \quad \text{(remainder 2)} \]
Thus, \( i^{442} = i^2 = -1 \).

### Step 5: Find \( i^{543} \)
Finally, we calculate \( 543 \mod 4 \):
\[ 543 \div 4 = 135 \quad \text{(remainder 3)} \]
Thus, \( i^{543} = i^3 = -i \).

### Step 6: Combine all the results
Now we can combine all the results:
\[ i^{248} + i^{341} + i^{442} + i^{543} = 1 + i - 1 - i \]
Simplifying this gives:
\[ 1 - 1 + i - i = 0 \]

### Final Answer
Thus, the value of \( i^{248} + i^{341} + i^{442} + i^{543} \) is \( \boxed{0} \).

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