Read the statements carefully and select the correct option.
Statement-1: If there are `(2n + 1)` terms in an A.P., then the ratio of the sum of odd terms and the sum of even terms is `(n + 1): n`.
Statement-II : If `S_(n)` the sum of first n terms of an A.P. is given by `S_(n)= 5n^2+ 3n` then its `n^(th)` term is `10n - 2`.

Options

A. Both Statement-I and Statement-II are true.
B. Both Statement-I and Statement-II are false.
C. Statement-I is true but Statement-II is false.
D. Statement-I is false but Statement-II is true.

Answer

To solve the question, we need to analyze both statements provided.

### Step 1: Analyze Statement-1
Statement-1 claims that if there are \( (2n + 1) \) terms in an A.P. (Arithmetic Progression), then the ratio of the sum of odd terms to the sum of even terms is \( (n + 1) : n \).

1. **Identify the terms in the A.P.**:
- The first term is \( a \) and the common difference is \( d \).
- The terms can be expressed as:
- Odd terms: \( a, a + 2d, a + 4d, \ldots \) (total \( n + 1 \) terms)
- Even terms: \( a + d, a + 3d, a + 5d, \ldots \) (total \( n \) terms)

2. **Calculate the sum of odd terms**:
- The sum of the first \( n + 1 \) odd terms:
\[ S_{\text{odd}} = \frac{(n + 1)}{2} \times (2a + 2nd) = (n + 1)(a + nd) \]

3. **Calculate the sum of even terms**:
- The sum of the first \( n \) even terms:
\[ S_{\text{even}} = \frac{n}{2} \times (2(a + d) + (n - 1) \cdot 2d) = n(a + d + (n - 1)d) = n(a + nd) \]

4. **Find the ratio of the sums**:
\[ \text{Ratio} = \frac{S_{\text{odd}}}{S_{\text{even}}} = \frac{(n + 1)(a + nd)}{n(a + nd)} = \frac{n + 1}{n} \]
This confirms that the ratio of the sum of odd terms to the sum of even terms is indeed \( (n + 1) : n \).

### Step 2: Analyze Statement-II
Statement-II states that if \( S_n = 5n^2 + 3n \), then the \( n^{th} \) term is \( 10n - 2 \).

1. **Find the \( n^{th} \) term**:
- The \( n^{th} \) term \( T_n \) can be found using the formula:
\[ T_n = S_n - S_{n-1} \]
- First, calculate \( S_{n-1} \):
\[ S_{n-1} = 5(n-1)^2 + 3(n-1) = 5(n^2 - 2n + 1) + 3n - 3 = 5n^2 - 10n + 5 + 3n - 3 = 5n^2 - 7n + 2 \]

2. **Calculate \( T_n \)**:
\[ T_n = S_n - S_{n-1} = (5n^2 + 3n) - (5n^2 - 7n + 2) = 3n + 7n - 2 = 10n - 2 \]
This confirms that the \( n^{th} \) term is indeed \( 10n - 2 \).

### Conclusion
Both statements are true. Therefore, the correct option is that both Statement-1 and Statement-2 are true.