If the 4th term of an A.P. is 12 and 12th term is 60, then the first term is

Options

A. (a) - 6
B. (b) - 2
C. (c) 3
D. (d) 5

Answer

To find the first term of the arithmetic progression (A.P.), we can follow these steps:

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

### Step 2: Set up the equations for the given terms.
We know:
- The 4th term \(A_4 = 12\)
- The 12th term \(A_{12} = 60\)

Using the formula:
1. For the 4th term:
\[ A_4 = a + (4 - 1)d = a + 3d = 12 \quad \text{(Equation 1)} \]

2. For the 12th term:
\[ A_{12} = a + (12 - 1)d = a + 11d = 60 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations simultaneously.
Now we have two equations:
1. \(a + 3d = 12\) (Equation 1)
2. \(a + 11d = 60\) (Equation 2)

We can subtract Equation 1 from Equation 2:
\[ (a + 11d) - (a + 3d) = 60 - 12 \]
This simplifies to:
\[ 11d - 3d = 48 \]
\[ 8d = 48 \]
\[ d = 6 \]

### Step 4: Substitute the value of \(d\) back into one of the equations to find \(a\).
Now, we can substitute \(d = 6\) back into Equation 1:
\[ a + 3(6) = 12 \]
\[ a + 18 = 12 \]
\[ a = 12 - 18 \]
\[ a = -6 \]

### Conclusion
The first term \(a\) of the A.P. is \(-6\).