The sum of `2^(nd)` and `7^(th)` term of an A.P. is 25 and that of `11^(th)` and `8^(th)` term is 75. Find the `2^(nd)` term

Options

A. 5
B. 0
C. 25
D. Can't be determined

Answer

To solve the problem, we need to use the properties of an Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding the Terms of A.P.**:
In an A.P., the nth term can be expressed as:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Setting Up the Equations**:
- The 2nd term \( a_2 \) is:
\[ a_2 = a + d \]
- The 7th term \( a_7 \) is:
\[ a_7 = a + 6d \]
- According to the problem, the sum of the 2nd and 7th terms is 25:
\[ a_2 + a_7 = (a + d) + (a + 6d) = 2a + 7d = 25 \quad \text{(1)} \]

- The 11th term \( a_{11} \) is:
\[ a_{11} = a + 10d \]
- The 8th term \( a_8 \) is:
\[ a_8 = a + 7d \]
- According to the problem, the sum of the 11th and 8th terms is 75:
\[ a_{11} + a_8 = (a + 10d) + (a + 7d) = 2a + 17d = 75 \quad \text{(2)} \]

3. **Solving the Equations**:
Now we have a system of two equations:
\[ 2a + 7d = 25 \quad \text{(1)} \]
\[ 2a + 17d = 75 \quad \text{(2)} \]

We can subtract equation (1) from equation (2):
\[ (2a + 17d) - (2a + 7d) = 75 - 25 \]
This simplifies to:
\[ 10d = 50 \]
Dividing both sides by 10 gives:
\[ d = 5 \]

4. **Finding the Value of \( a \)**:
Now substitute \( d = 5 \) back into equation (1):
\[ 2a + 7(5) = 25 \]
Simplifying this gives:
\[ 2a + 35 = 25 \]
Subtracting 35 from both sides:
\[ 2a = 25 - 35 \]
\[ 2a = -10 \]
Dividing both sides by 2:
\[ a = -5 \]

5. **Finding the 2nd Term**:
Now we can find the 2nd term \( a_2 \):
\[ a_2 = a + d = -5 + 5 = 0 \]

### Final Answer:
The 2nd term of the A.P. is \( \boxed{0} \).