Prove that `i^(n)+i^(n+1)+i^(n+2)+i^(n+3)=0` for all `n inN`

Answer

To prove that \( i^n + i^{n+1} + i^{n+2} + i^{n+3} = 0 \) for all \( n \in \mathbb{N} \), we will follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ i^n + i^{n+1} + i^{n+2} + i^{n+3} \]

### Step 2: Factor out \( i^n \)
We can factor \( i^n \) out of the expression:
\[ i^n (1 + i + i^2 + i^3) \]

### Step 3: Calculate the powers of \( i \)
Now, we need to evaluate \( 1 + i + i^2 + i^3 \):
- \( i^0 = 1 \)
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)

### Step 4: Substitute the values
Substituting these values into the expression, we get:
\[ 1 + i - 1 - i \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ 1 - 1 + i - i = 0 \]

### Step 6: Final expression
Thus, we have:
\[ i^n (1 + i + i^2 + i^3) = i^n \cdot 0 = 0 \]

### Conclusion
Therefore, we conclude that:
\[ i^n + i^{n+1} + i^{n+2} + i^{n+3} = 0 \]
for all \( n \in \mathbb{N} \).