Question:

Step 1: Understand the Problem's Core The problem concerning an Arithmetic Progression (AP) fundamentally revolves around two key quantities: its first term and its common difference. Recognizing these as the core elements is the initial step in formulating a solution. Step 2: Parameterize the Progression and Formulate Conditions Algebraically To address the problem systematically, the recommended approach is to parameterize the Arithmetic Progression using its first term, denoted as $a_1$, and its common difference, denoted as $d$. Subsequently, every condition provided in the problem statement is translated into algebraic equations or expressions involving these parameters. Step 3: Apply Standard Arithmetic Progression Formulas Once the conditions are expressed in algebraic terms, the next step involves invoking the standard formulas associated with Arithmetic Progressions. These commonly include: The formula for the $n$-th term: $$a_n = a_1 + (n-1)d$$ The formula for the sum of the first $n$ terms: $$S_n = \frac{n}{2}(2a_1 + (n-1)d) \quad \text{or} \quad S_n = \frac{n}{2}(a_1 + a_n)$$ These formulas are used to set up a system of equations based on the algebraic conditions derived in the previous step. Step 4: Simplify and Solve Algebraically With the equations established using the standard AP formulas, the process continues by simplifying these algebraic expressions step-by-step. This often involves applying various algebraic techniques such as substitution, elimination, or other methods to solve for the unknown variables ($a_1$, $d$, or any other quantity required by the problem). Step 5: Conclude the Solution After executing the necessary algebraic manipulations and computations to determine the required value, the final answer is confirmed. The final answer matches Option 4.