Suppose n is a natural number such that `|i + 2i^2 + 3i^3 +...... + ni^n|=18sqrt2` where `i` is the square root of `-1`. Then n is

Options

A. `9`
B. `18`
C. `36`
D. `72`

Explanation

To solve the problem, we need to evaluate the expression \( |i + 2i^2 + 3i^3 + \ldots + ni^n| = 18\sqrt{2} \) and find the natural number \( n \).

### Step-by-Step Solution:

1. **Evaluate the powers of \( i \)**:
- Recall that \( i = \sqrt{-1} \).
- The powers of \( i \) cycle every four terms:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \), and so on.

2. **Break down the series**:
- The expression can be rewritten as:
\[ S = i + 2(-1) + 3(-i) + 4(1) + 5(i) + 6(-1) + 7(-i) + 8(1) + \ldots + n(i^n) \]
- Group the terms based on their real and imaginary parts.

3. **Identify real and imaginary parts**:
- The real part consists of terms where \( i^k = 1 \) or \( i^k = -1 \):
- For even \( k \): \( k \) contributes to the real part.
- For odd \( k \): \( k \) contributes negatively if \( k \equiv 2 \mod 4 \) and positively if \( k \equiv 0 \mod 4 \).
- The imaginary part consists of terms where \( i^k = i \) or \( i^k = -i \):
- For odd \( k \): \( k \) contributes to the imaginary part.

4. **Sum the real parts**:
- The real parts can be summed as:
\[ \text{Real part} = (4 + 8 + \ldots) - (2 + 6 + \ldots) \]
- The count of terms in each series depends on \( n \).

5. **Sum the imaginary parts**:
- The imaginary parts can be summed as:
\[ \text{Imaginary part} = (1 + 5 + \ldots) - (3 + 7 + \ldots) \]

6. **Calculate the total contributions**:
- Let \( n \) be even, say \( n = 2k \):
- Real part: \( 2 \left(1 + 2 + \ldots + k\right) = k(k + 1) \)
- Imaginary part: \( k \) terms contribute positively and negatively.

7. **Combine real and imaginary parts**:
- The total can be expressed as:
\[ S = \text{Real part} + i \cdot \text{Imaginary part} \]

8. **Find the modulus**:
- The modulus is given by:
\[ |S| = \sqrt{(\text{Real part})^2 + (\text{Imaginary part})^2} \]
- Set this equal to \( 18\sqrt{2} \).

9. **Solve for \( n \)**:
- From the modulus equation, solve for \( n \):
\[ \sqrt{\left(\frac{n}{2}\right)^2 + \left(-\frac{n}{2}\right)^2} = 18\sqrt{2} \]
- This simplifies to:
\[ \frac{n}{\sqrt{2}} = 18\sqrt{2} \implies n = 36 \]

### Final Answer:
Thus, the value of \( n \) is \( \boxed{36} \).