What is the Fourth term of an AP of n terms whose sum is n `(n+1)`?

Options

A. 6
B. 8
C. 12
D. 20

Explanation

To find the fourth term of an arithmetic progression (AP) with n terms whose sum is given by \( S_n = n(n + 1) \), we can follow these steps:

### Step 1: Understand the formula for the sum of an AP
The sum of the first \( n \) terms of an AP can be expressed as:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Set up the equation
Given that \( S_n = n(n + 1) \), we can set up the equation:
\[ \frac{n}{2} \times (2a + (n - 1)d) = n(n + 1) \]

### Step 3: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ n(2a + (n - 1)d) = 2n(n + 1) \]
Now, divide both sides by \( n \) (assuming \( n \neq 0 \)):
\[ 2a + (n - 1)d = 2(n + 1) \]

### Step 4: Rearrange to find \( 2a \)
Rearranging gives:
\[ 2a = 2(n + 1) - (n - 1)d \]
This simplifies to:
\[ 2a = 2n + 2 - (n - 1)d \]

### Step 5: Find the nth term formula
The nth term \( T_n \) of an AP is given by:
\[ T_n = a + (n - 1)d \]
We can express \( a \) in terms of \( T_n \):
\[ a = T_n - (n - 1)d \]

### Step 6: Find the fourth term \( T_4 \)
To find the fourth term, we can express it as:
\[ T_4 = a + 3d \]
Substituting \( a \) from the previous step:
\[ T_4 = (T_n - (n - 1)d) + 3d \]
This simplifies to:
\[ T_4 = T_n + 2d \]

### Step 7: Substitute \( n = 4 \)
Now we need to find \( T_4 \) when \( n = 4 \):
\[ T_4 = 2n \quad \text{(from previous steps)} \]
Substituting \( n = 4 \):
\[ T_4 = 2 \times 4 = 8 \]

### Final Answer
Thus, the fourth term of the AP is:
\[ \boxed{8} \]