Find the common difference of an AP whose `n^(th)` term is 6n + 2.

Answer

To find the common difference of an arithmetic progression (AP) whose \( n^{th} \) term is given by \( T_n = 6n + 2 \), we can follow these steps:

### Step 1: Write the expression for the \( n^{th} \) term
The \( n^{th} \) term of the AP is given as:
\[ T_n = 6n + 2 \]

### Step 2: Write the expression for the \( (n-1)^{th} \) term
To find the common difference, we also need the \( (n-1)^{th} \) term:
\[ T_{n-1} = 6(n-1) + 2 \]
Expanding this, we get:
\[ T_{n-1} = 6n - 6 + 2 = 6n - 4 \]

### Step 3: Find the common difference
The common difference \( d \) of an AP is defined as:
\[ d = T_n - T_{n-1} \]
Substituting the expressions we found:
\[ d = (6n + 2) - (6n - 4) \]
Now, simplifying this:
\[ d = 6n + 2 - 6n + 4 \]
The \( 6n \) terms cancel out:
\[ d = 2 + 4 = 6 \]

### Conclusion
Thus, the common difference of the AP is:
\[ \boxed{6} \]