The 12th term of an A.P. is 14 more than the 5th term. The sum of these terms is 36. Find the A.P.

Explanation

To solve the problem step by step, we will denote the first term of the Arithmetic Progression (A.P.) as \( a \) and the common difference as \( d \).

### Step 1: Set up the equations based on the problem statement
We know that:
- The 12th term of the A.P. is given by \( A_{12} = a + 11d \).
- The 5th term of the A.P. is given by \( A_{5} = a + 4d \).
- According to the problem, \( A_{12} \) is 14 more than \( A_{5} \).

This gives us the equation:
\[ a + 11d = (a + 4d) + 14 \]

### Step 2: Simplify the equation
Now, we can simplify the equation:
\[ a + 11d = a + 4d + 14 \]
Subtract \( a \) from both sides:
\[ 11d = 4d + 14 \]
Now, subtract \( 4d \) from both sides:
\[ 11d - 4d = 14 \]
This simplifies to:
\[ 7d = 14 \]

### Step 3: Solve for \( d \)
Now, divide both sides by 7:
\[ d = 2 \]

### Step 4: Use the sum of the terms
We are also given that the sum of the 12th and 5th terms is 36:
\[ A_{12} + A_{5} = 36 \]
Substituting the expressions for \( A_{12} \) and \( A_{5} \):
\[ (a + 11d) + (a + 4d) = 36 \]
Combine like terms:
\[ 2a + 15d = 36 \]

### Step 5: Substitute \( d \) into the equation
Now substitute \( d = 2 \) into the equation:
\[ 2a + 15(2) = 36 \]
This simplifies to:
\[ 2a + 30 = 36 \]

### Step 6: Solve for \( a \)
Now, subtract 30 from both sides:
\[ 2a = 36 - 30 \]
This simplifies to:
\[ 2a = 6 \]
Now, divide both sides by 2:
\[ a = 3 \]

### Step 7: Write the A.P.
Now that we have \( a = 3 \) and \( d = 2 \), we can write the terms of the A.P.:
- First term: \( a = 3 \)
- Second term: \( a + d = 3 + 2 = 5 \)
- Third term: \( a + 2d = 3 + 2 \times 2 = 7 \)
- Fourth term: \( a + 3d = 3 + 3 \times 2 = 9 \)

Thus, the A.P. is:
\[ 3, 5, 7, 9, \ldots \]

### Final Answer:
The A.P. is \( 3, 5, 7, 9, \ldots \)

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