The third term of an A.P. is `25` and the tenth term is `-3`. find the first term and the common difference.

Answer

To solve the problem, we need to use the formula for the nth term of an arithmetic progression (A.P.), which is given by:

\[ a_n = a + (n-1)d \]

where:
- \( a_n \) is the nth term,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

### Step 1: Set up the equations based on the given terms.
From the problem, we know:
- The third term (\( a_3 \)) is 25:
\[ a + 2d = 25 \] (Equation 1)

- The tenth term (\( a_{10} \)) is -3:
\[ a + 9d = -3 \] (Equation 2)

### Step 2: Solve the equations simultaneously.
We have two equations:
1. \( a + 2d = 25 \) (Equation 1)
2. \( a + 9d = -3 \) (Equation 2)

To eliminate \( a \), we can subtract Equation 1 from Equation 2:

\[ (a + 9d) - (a + 2d) = -3 - 25 \]

This simplifies to:

\[ 7d = -28 \]

### Step 3: Solve for the common difference \( d \).
Dividing both sides by 7 gives us:

\[ d = -4 \]

### Step 4: Substitute \( d \) back into one of the equations to find \( a \).
Now we can substitute \( d = -4 \) back into Equation 1 to find \( a \):

\[ a + 2(-4) = 25 \]

This simplifies to:

\[ a - 8 = 25 \]

Adding 8 to both sides gives:

\[ a = 33 \]

### Conclusion:
The first term \( a \) is 33 and the common difference \( d \) is -4.

### Final Answer:
- First term \( a = 33 \)
- Common difference \( d = -4 \)