If `(x + 1), 3x and (4x + 2)` are first three terms of an A.P. then its `5^(th)` term is

Options

A. 14
B. 19
C. 24
D. 28

Explanation

To solve the problem step by step, we will follow the properties of an Arithmetic Progression (A.P.).

### Step 1: Identify the terms
The first three terms of the A.P. are given as:
- First term (a) = \( x + 1 \)
- Second term (b) = \( 3x \)
- Third term (c) = \( 4x + 2 \)

### Step 2: Use the property of A.P.
In an A.P., the difference between consecutive terms is constant. Therefore, we can set up the equation:
\[ b - a = c - b \]
Substituting the terms:
\[ 3x - (x + 1) = (4x + 2) - 3x \]

### Step 3: Simplify the equation
Now, simplify both sides:
- Left side:
\[ 3x - x - 1 = 2x - 1 \]
- Right side:
\[ 4x + 2 - 3x = x + 2 \]
So, we have:
\[ 2x - 1 = x + 2 \]

### Step 4: Solve for \( x \)
Now, isolate \( x \):
\[ 2x - x = 2 + 1 \]
\[ x = 3 \]

### Step 5: Find the first term and common difference
Now that we have \( x \), we can find the first term and the common difference:
- First term \( a = x + 1 = 3 + 1 = 4 \)
- Second term \( b = 3x = 3 \times 3 = 9 \)
- Third term \( c = 4x + 2 = 4 \times 3 + 2 = 12 + 2 = 14 \)

Now, calculate the common difference \( d \):
\[ d = b - a = 9 - 4 = 5 \]

### Step 6: Find the fifth term
The formula for the \( n^{th} \) term of an A.P. is given by:
\[ T_n = a + (n - 1)d \]
For the fifth term (\( n = 5 \)):
\[ T_5 = 4 + (5 - 1) \times 5 \]
\[ T_5 = 4 + 4 \times 5 \]
\[ T_5 = 4 + 20 = 24 \]

### Final Answer
The fifth term of the A.P. is \( \boxed{24} \).