If the 3rd and 7th terms of an A.P. are 17 and 27 respectively . Find the first term of the A.P.:

Options

A. 9
B. 12
C. 14
D. none of these

Answer

To find the first term of the arithmetic progression (A.P.) given that the 3rd term is 17 and the 7th term is 27, we can follow these steps:

### Step 1: Write the formulas for the 3rd and 7th terms of the A.P.
The n-th term of an A.P. can be expressed as:
\[ T_n = a + (n - 1)d \]
where \( a \) is the first term and \( d \) is the common difference.

For the 3rd term (\( T_3 \)):
\[ T_3 = a + 2d = 17 \]
This is our **first equation**.

For the 7th term (\( T_7 \)):
\[ T_7 = a + 6d = 27 \]
This is our **second equation**.

### Step 2: Set up the equations
We have the following two equations:
1. \( a + 2d = 17 \) (Equation 1)
2. \( a + 6d = 27 \) (Equation 2)

### Step 3: Subtract the first equation from the second
To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (a + 6d) - (a + 2d) = 27 - 17 \]
This simplifies to:
\[ 6d - 2d = 10 \]
\[ 4d = 10 \]

### Step 4: Solve for \( d \)
Now, divide both sides by 4:
\[ d = \frac{10}{4} = \frac{5}{2} \]

### Step 5: Substitute \( d \) back into one of the equations to find \( a \)
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 2d = 17 \]
Substituting \( d = \frac{5}{2} \):
\[ a + 2 \left(\frac{5}{2}\right) = 17 \]
This simplifies to:
\[ a + 5 = 17 \]

### Step 6: Solve for \( a \)
Now, subtract 5 from both sides:
\[ a = 17 - 5 = 12 \]

### Conclusion
The first term of the A.P. is:
\[ a = 12 \]