An A.P. consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term of the A.P.

Answer

To find the 29th term of the arithmetic progression (A.P.) with the given conditions, we can follow these steps:

### Step 1: Identify the given values
We know:
- The number of terms (n) = 50
- The 3rd term (a₃) = 12
- The last term (a₅₀) = 106

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
where:
- \( a \) = first term
- \( d \) = common difference

### Step 3: Set up equations based on the given terms
From the last term:
\[ a_{50} = a + (50 - 1) \cdot d = 106 \]
This simplifies to:
\[ a + 49d = 106 \] (Equation 1)

From the 3rd term:
\[ a_{3} = a + (3 - 1) \cdot d = 12 \]
This simplifies to:
\[ a + 2d = 12 \] (Equation 2)

### Step 4: Solve the equations
Now, we have two equations:
1. \( a + 49d = 106 \) (Equation 1)
2. \( a + 2d = 12 \) (Equation 2)

We can subtract Equation 2 from Equation 1:
\[ (a + 49d) - (a + 2d) = 106 - 12 \]
This simplifies to:
\[ 49d - 2d = 94 \]
\[ 47d = 94 \]
Now, divide both sides by 47:
\[ d = \frac{94}{47} = 2 \]

### Step 5: Find the first term (a)
Now that we have \( d = 2 \), we can substitute this value back into Equation 2 to find \( a \):
\[ a + 2(2) = 12 \]
\[ a + 4 = 12 \]
Subtract 4 from both sides:
\[ a = 12 - 4 = 8 \]

### Step 6: Find the 29th term
Now we can find the 29th term using the formula:
\[ a_{29} = a + (29 - 1) \cdot d \]
Substituting the values of \( a \) and \( d \):
\[ a_{29} = 8 + (28) \cdot 2 \]
\[ a_{29} = 8 + 56 = 64 \]

### Final Answer
Thus, the 29th term of the A.P. is:
\[ \boxed{64} \]