The sum of the coefficient in the expansion of `(1+ax-2x^(2))^(n)` is

Options

A. positive, when `a lt 1` and `n = 2k, k in N`
B. negative, when `a lt 1` and `n = 2k + 1 , k in N`
C. positive, when `a gt 1` and `n in N`
D. zero, when `a = 1`

Explanation

To find the sum of the coefficients in the expansion of \((1 + ax - 2x^2)^n\), we can follow these steps:

### Step 1: Understand the Problem
We need to find the sum of the coefficients of the polynomial when expanded. The sum of the coefficients of a polynomial \(P(x)\) can be found by evaluating \(P(1)\).

### Step 2: Substitute \(x = 1\)
We substitute \(x = 1\) into the expression:
\[ P(1) = (1 + a(1) - 2(1)^2)^n \]
This simplifies to:
\[ P(1) = (1 + a - 2)^n \]
\[ P(1) = (a - 1)^n \]

### Step 3: Analyze the Result
The expression \((a - 1)^n\) gives us the sum of the coefficients in the expansion.

### Step 4: Consider Different Cases
Now we can analyze the expression \((a - 1)^n\) based on the value of \(a\):
1. If \(a < 1\), then \(a - 1 < 0\) and \((a - 1)^n\) will be positive if \(n\) is even, and negative if \(n\) is odd.
2. If \(a = 1\), then \((a - 1)^n = 0^n = 0\).
3. If \(a > 1\), then \(a - 1 > 0\) and \((a - 1)^n\) will always be positive.

### Conclusion
Thus, the sum of the coefficients in the expansion of \((1 + ax - 2x^2)^n\) is given by:
\[ (a - 1)^n \]