Show that : `i^(101)+i^(102)+i^(103)+i^(104)=0`

Answer

To show that \( i^{101} + i^{102} + i^{103} + i^{104} = 0 \), we can follow these steps:

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

### Step 2: Calculate each term
Now, we will calculate \( i^{101} \), \( i^{102} \), \( i^{103} \), and \( i^{104} \) using the cyclic property.

- To find \( i^{101} \):
\[ 101 \mod 4 = 1 \quad \Rightarrow \quad i^{101} = i \]

- To find \( i^{102} \):
\[ 102 \mod 4 = 2 \quad \Rightarrow \quad i^{102} = -1 \]

- To find \( i^{103} \):
\[ 103 \mod 4 = 3 \quad \Rightarrow \quad i^{103} = -i \]

- To find \( i^{104} \):
\[ 104 \mod 4 = 0 \quad \Rightarrow \quad i^{104} = 1 \]

### Step 3: Substitute back into the equation
Now substitute these values back into the original expression:
\[ i^{101} + i^{102} + i^{103} + i^{104} = i + (-1) + (-i) + 1 \]

### Step 4: Simplify the expression
Now, we can simplify:
\[ i - i - 1 + 1 = 0 \]

### Conclusion
Thus, we have shown that:
\[ i^{101} + i^{102} + i^{103} + i^{104} = 0 \]

This proves the statement.

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