Arithmetic Progression

Given the sum of the first n terms of an arithmetic progression (AP): , we need to determine the first term and the common difference .

Step 1: Use Formulas for Sum of AP

The sum of the first terms of an AP is given by:

Equating the given expression for :

Multiplying both sides by 2 to clear the fraction gives:

Simplifying, divide by :

Step 2: Find and

Setting to find :

For :

Solution: The first term is 8, and the common difference is 6.

This matches the option .

To solve the problem, we will use the formula for the nth term of an arithmetic progression (AP):

where is the first term, is the common difference, and is the nth term.

Given:

• The 5th term ( ) is 20:
• The 12th term ( ) is 41:

We have two equations:

Subtract the first equation from the second to eliminate :

Solve for :

Substitute back into the first equation to find :

Solving for :

Therefore, the first term of the AP is 8. The correct answer is: 8