Statement-1: If `underset(r=1)overset(n)(sum)r^(3)((.^(n)C_(r))/(.^(n)C_(r-1)))^(2)=196`, then the sum of the coeficients of powerr of xin the expansion of the polynomial `(x-3x^(2)+x^(3))^(n)` is -1. Statement-2: `(.^(n)C_(r))/(.^(n)C_(r-1))=(n-r+1)/(r) AA n in N and r in W`.

Answer

To solve the given problem, we will follow the steps outlined in the video transcript, breaking it down into a structured format.

### Step 1: Understand the Given Statement
We need to evaluate the statement:
\[ \sum_{r=1}^{n} r^3 \left( \frac{nC_r}{nC_{r-1}} \right)^2 = 196 \]
We also need to verify the second statement:
\[ \frac{nC_r}{nC_{r-1}} = \frac{n - r + 1}{r} \]

### Step 2: Simplify the Binomial Coefficient Ratio
Using the formula for binomial coefficients:
\[ nC_r = \frac{n!}{r!(n-r)!} \]
we can express \(nC_{r-1}\):
\[ nC_{r-1} = \frac{n!}{(r-1)!(n-r+1)!} \]
Thus,
\[ \frac{nC_r}{nC_{r-1}} = \frac{n!}{r!(n-r)!} \cdot \frac{(r-1)!(n-r+1)!}{n!} = \frac{(n-r+1)}{r} \]
This confirms that the second statement is correct.

### Step 3: Substitute into the Summation
Now, substituting the expression we derived into the summation:
\[ \sum_{r=1}^{n} r^3 \left( \frac{n - r + 1}{r} \right)^2 = 196 \]
This simplifies to:
\[ \sum_{r=1}^{n} r^3 \cdot \frac{(n - r + 1)^2}{r^2} = 196 \]
This can be rewritten as:
\[ \sum_{r=1}^{n} r(n - r + 1)^2 = 196 \]

### Step 4: Expand the Summation
Expanding \((n - r + 1)^2\):
\[ (n - r + 1)^2 = n^2 - 2nr + 2n + r^2 - 2r + 1 \]
Thus,
\[ \sum_{r=1}^{n} r(n - r + 1)^2 = \sum_{r=1}^{n} r(n^2 - 2nr + 2n + r^2 - 2r + 1) \]
This can be separated into individual summations:
\[ n^2 \sum_{r=1}^{n} r - 2n \sum_{r=1}^{n} r^2 + 2n \sum_{r=1}^{n} r + \sum_{r=1}^{n} r^3 - 2 \sum_{r=1}^{n} r^2 + \sum_{r=1}^{n} r \]

### Step 5: Use Known Summation Formulas
Using the formulas:
- \(\sum_{r=1}^{n} r = \frac{n(n+1)}{2}\)
- \(\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}\)
- \(\sum_{r=1}^{n} r^3 = \left( \frac{n(n+1)}{2} \right)^2\)

We can substitute these values into the equation and solve for \(n\).

### Step 6: Solve for \(n\)
After substituting and simplifying, we find:
\[ \text{(expression in terms of n)} = 196 \]
Solving this will yield \(n = 6\).

### Step 7: Find the Coefficient of \(x\) in the Polynomial
Now, substitute \(n = 6\) into the polynomial:
\[ (x - 3x^2 + x^3)^6 \]
To find the sum of the coefficients, substitute \(x = 1\):
\[ (1 - 3 + 1)^6 = (-1)^6 = 1 \]

### Conclusion
The sum of the coefficients of the power of \(x\) in the expansion is \(1\), which contradicts the statement that it is \(-1\). Therefore, Statement 1 is incorrect, while Statement 2 is correct.