In an A.P. , 3 is the first term. If the second, tenth and thirty fourth terms forms a G.P., then the fourth term of the A.P. is

Options

A. 2
B. 1
C. 4
D. 6

Answer

To solve the problem step by step, we will follow the logic outlined in the video transcript.

### Step 1: Identify the first term and the formula for the nth term of an A.P.
The first term \( a \) of the arithmetic progression (A.P.) is given as 3. The formula for the nth term of an A.P. is:
\[ A_n = a + (n-1)d \]
where \( d \) is the common difference.

### Step 2: Write the expressions for the 2nd, 10th, and 34th terms
Using the formula for the nth term, we can express the 2nd, 10th, and 34th terms as follows:

- **2nd term** (\( A_2 \)):
\[ A_2 = a + (2-1)d = 3 + d \]

- **10th term** (\( A_{10} \)):
\[ A_{10} = a + (10-1)d = 3 + 9d \]

- **34th term** (\( A_{34} \)):
\[ A_{34} = a + (34-1)d = 3 + 33d \]

### Step 3: Set up the condition for the terms to form a G.P.
The terms \( A_2 \), \( A_{10} \), and \( A_{34} \) form a geometric progression (G.P.). For three terms \( m, n, p \) to be in G.P., the condition is:
\[ n^2 = m \cdot p \]
Applying this to our terms:
\[ (3 + 9d)^2 = (3 + d)(3 + 33d) \]

### Step 4: Expand both sides of the equation
**Left-hand side**:
\[ (3 + 9d)^2 = 9 + 81d^2 + 54d \]

**Right-hand side**:
\[ (3 + d)(3 + 33d) = 9 + 99d + 3d + 33d^2 = 9 + 102d + 33d^2 \]

### Step 5: Set the left-hand side equal to the right-hand side
Now we have:
\[ 9 + 81d^2 + 54d = 9 + 102d + 33d^2 \]

### Step 6: Simplify the equation
Subtract \( 9 \) from both sides:
\[ 81d^2 + 54d = 102d + 33d^2 \]
Rearranging gives:
\[ 81d^2 - 33d^2 + 54d - 102d = 0 \]
This simplifies to:
\[ 48d^2 - 48d = 0 \]

### Step 7: Factor the equation
Factoring out \( 48d \):
\[ 48d(d - 1) = 0 \]
This gives us two solutions:
\[ d = 0 \quad \text{or} \quad d = 1 \]

### Step 8: Determine the value of \( d \)
Since \( d = 0 \) would mean all terms are the same (not forming a G.P.), we take:
\[ d = 1 \]

### Step 9: Calculate the 4th term of the A.P.
Now we can find the 4th term \( A_4 \):
\[ A_4 = a + (4-1)d = 3 + 3 \cdot 1 = 3 + 3 = 6 \]

### Final Answer
The fourth term of the A.P. is:
\[ \boxed{6} \]