For any positive integer n, find the value of `i^(n)+i^(n+1)+i^(n+2)+i^(n+3)+i^(n+4)+i^(n+5)+i^(n+6)+i^(n+7)`.

Answer

To solve the problem of finding the value of \( i^{n} + i^{n+1} + i^{n+2} + i^{n+3} + i^{n+4} + i^{n+5} + i^{n+6} + i^{n+7} \), we can follow these steps:

### Step 1: Factor out \( i^n \)
We start by factoring out \( i^n \) from the expression:
\[ i^{n} + i^{n+1} + i^{n+2} + i^{n+3} + i^{n+4} + i^{n+5} + i^{n+6} + i^{n+7} = i^n (1 + i + i^2 + i^3 + i^4 + i^5 + i^6 + i^7) \]

**Hint:** Look for common factors in the terms to simplify the expression.

### Step 2: Identify the powers of \( i \)
Recall the powers of \( i \):
- \( i^0 = 1 \)
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \) (and this pattern repeats every 4 terms)

Using this, we can express \( i^4, i^5, i^6, \) and \( i^7 \) in terms of \( i^0, i^1, i^2, \) and \( i^3 \):
- \( i^4 = 1 \)
- \( i^5 = i \)
- \( i^6 = -1 \)
- \( i^7 = -i \)

### Step 3: Substitute the values into the expression
Now substitute these values back into the expression:
\[ 1 + i + (-1) + (-i) + 1 + i + (-1) + (-i) \]

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

### Step 5: Combine with the factored term
Thus, we have:
\[ i^n (1 + i + i^2 + i^3 + i^4 + i^5 + i^6 + i^7) = i^n \cdot 0 = 0 \]

### Final Result
The value of \( i^{n} + i^{n+1} + i^{n+2} + i^{n+3} + i^{n+4} + i^{n+5} + i^{n+6} + i^{n+7} \) is:
\[ \boxed{0} \]