Satement 1 If the nth termn of a series is `2n^(3)+3n^(2)-4`, then the second order differences must be an AP.
Statement 2 If nth term of a series is a polynomial of degree m, then mth order differences of series are constant.

Options

A. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1
B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1
C. Statement 1 is true, Statement 2 is false
D. Statement 1 is false, Statement 2 is true

Explanation

To solve the problem, we need to analyze the two statements given:

**Statement 1:** If the nth term of a series is \(2n^3 + 3n^2 - 4\), then the second order differences must be an AP (Arithmetic Progression).

**Statement 2:** If the nth term of a series is a polynomial of degree \(m\), then the \(m\)th order differences of the series are constant.

### Step-by-Step Solution:

1. **Identify the nth term:**
The nth term of the series is given as:
\[ T(n) = 2n^3 + 3n^2 - 4 \]

2. **Calculate the first few terms of the series:**
We will calculate the first five terms by substituting \(n = 1, 2, 3, 4, 5\):
- For \(n = 1\):
\[ T(1) = 2(1)^3 + 3(1)^2 - 4 = 2 + 3 - 4 = 1 \]
- For \(n = 2\):
\[ T(2) = 2(2)^3 + 3(2)^2 - 4 = 2(8) + 3(4) - 4 = 16 + 12 - 4 = 24 \]
- For \(n = 3\):
\[ T(3) = 2(3)^3 + 3(3)^2 - 4 = 2(27) + 3(9) - 4 = 54 + 27 - 4 = 77 \]
- For \(n = 4\):
\[ T(4) = 2(4)^3 + 3(4)^2 - 4 = 2(64) + 3(16) - 4 = 128 + 48 - 4 = 172 \]
- For \(n = 5\):
\[ T(5) = 2(5)^3 + 3(5)^2 - 4 = 2(125) + 3(25) - 4 = 250 + 75 - 4 = 321 \]

The first five terms are:
\[ T(1) = 1, \quad T(2) = 24, \quad T(3) = 77, \quad T(4) = 172, \quad T(5) = 321 \]

3. **Calculate the first order differences:**
The first order differences are calculated as follows:
\[ D_1(1) = T(2) - T(1) = 24 - 1 = 23 \]
\[ D_1(2) = T(3) - T(2) = 77 - 24 = 53 \]
\[ D_1(3) = T(4) - T(3) = 172 - 77 = 95 \]
\[ D_1(4) = T(5) - T(4) = 321 - 172 = 149 \]
The first order differences are:
\[ D_1 = 23, \quad 53, \quad 95, \quad 149 \]

4. **Calculate the second order differences:**
The second order differences are calculated as follows:
\[ D_2(1) = D_1(2) - D_1(1) = 53 - 23 = 30 \]
\[ D_2(2) = D_1(3) - D_1(2) = 95 - 53 = 42 \]
\[ D_2(3) = D_1(4) - D_1(3) = 149 - 95 = 54 \]
The second order differences are:
\[ D_2 = 30, \quad 42, \quad 54 \]

5. **Check if the second order differences are in AP:**
To check if the second order differences are in AP, we calculate the differences between them:
\[ D_2(2) - D_2(1) = 42 - 30 = 12 \]
\[ D_2(3) - D_2(2) = 54 - 42 = 12 \]
Since the differences are constant (both equal to 12), the second order differences are indeed in AP.

### Conclusion:
- **Statement 1 is True**: The second order differences are in AP.
- **Statement 2 is True**: The \(m\)th order differences of a polynomial of degree \(m\) are constant.

Thus, both statements are true, and Statement 2 provides a correct explanation for Statement 1.