If the sum of `n` terms of an A.P. be `3n^2-n` and its common difference is 6, then its first term is

Options

A. `2`
B. `3`
C. `1`
D. `4`

Answer

To solve the problem step by step, we will follow the mathematical approach based on the information provided.

### Step 1: Understand the given information
We are given that the sum of the first `n` terms of an Arithmetic Progression (A.P.) is given by:
\[ S_n = 3n^2 - n \]
And the common difference \( d = 6 \).

### Step 2: Use the formula for the sum of the first `n` terms of an A.P.
The formula for the sum of the first `n` terms of an A.P. is:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Where \( a \) is the first term.

### Step 3: Substitute the known values into the formula
Substituting \( S_n \) and \( d \) into the formula, we have:
\[ 3n^2 - n = \frac{n}{2} \times (2a + (n - 1) \cdot 6) \]

### Step 4: Simplify the equation
This simplifies to:
\[ 3n^2 - n = \frac{n}{2} \times (2a + 6n - 6) \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 2(3n^2 - n) = n(2a + 6n - 6) \]
This gives us:
\[ 6n^2 - 2n = n(2a + 6n - 6) \]

### Step 5: Divide both sides by \( n \) (assuming \( n \neq 0 \))
\[ 6n - 2 = 2a + 6n - 6 \]

### Step 6: Rearrange the equation to solve for \( a \)
Now, we can rearrange the equation:
\[ 6n - 2 - 6n + 6 = 2a \]
This simplifies to:
\[ 4 = 2a \]

### Step 7: Solve for \( a \)
Dividing both sides by 2:
\[ a = 2 \]

### Final Answer
The first term \( a \) of the A.P. is:
\[ \boxed{2} \]

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