The nth term of an A. P. is \(\mathrm{p}^{2}\) and the sum of the first \(\mathrm{n}\) terms is \(\mathrm{s}^{2}\) The first term is (A) \(\left[\left(\mathrm{p}^{2} \mathrm{n}+2 \mathrm{~s}^{2}\right) / \mathrm{n}\right]\) (B) \(\left[\left(2 \mathrm{~s}^{2}+\mathrm{p}^{2} \mathrm{n}\right) / \mathrm{n}^{2}\right]\) (C) \(\left[\left(\mathrm{ps}^{2}-\mathrm{p}^{2} \mathrm{~s}\right) / \mathrm{n}\right]\) (D) \(\left[\left(2 \mathrm{~s}^{2}-\mathrm{p}^{2} \mathrm{n}\right) / \mathrm{n}\right]\)

Understanding the nth term of an arithmetic progression (AP) is essential to grasp the behavior of the sequence as it progresses. The nth term is based on a pattern that starts with the first term and follows with each subsequent term having a constant addition, known as the common difference.

An AP is defined by two parameters - the first term, often denoted as 'a', and the common difference, denoted as 'd'. The nth term, i.e., the term in the nth position of the sequence, is then given by the straightforward linear formula:
\[T_n = a + (n-1)d\].
This elegantly encapsulates the idea that each term is a specific number of increments (common differences) away from the first term. For students, visualizing this on a number line or plotting the first several terms can be very enlightening and aid in fully grasping this concept. It’s also crucial in solving problems related to finding specific terms within the sequence.

Moving from individual terms to considering a collection of terms, the sum of the first n terms of an AP is a common calculation in algebra. The formula to find this sum is also elegantly simple:
\[S_n = \frac{n}{2}(2a + (n-1)d)\].
Here, \(S_n\) represents the sum of the sequence up to the nth term, 'n' is the number of terms, 'a' is the first term, and 'd' is the common difference. This formula essentially calculates the average of the first and nth term and multiplies it by the number of terms.

This concept is similar to finding the area of a trapezoid, where the parallel sides represent the first and the nth terms, and the height represents the number of terms.

Understanding various AP formulas is essential in solving a myriad of problems involving arithmetic progressions. Here is a quick overview of the key formulas:

• The nth term of an AP: \(T_n = a + (n-1)d\)
• The sum of the first n terms of an AP: \(S_n = \frac{n}{2}(2a + (n-1)d)\)
• The sum of an AP when the last term is known: \(S_n = \frac{n}{2}(a + l)\), where 'l' is the last term
• The average term (arithmetic mean) of the first n terms: \(A_n = \frac{S_n}{n}\)

These formulas play a pivotal role in solving numerous real-world and theoretical problems. It is important to familiarize oneself with the conditions under which each formula is applicable and the relationships between the variables involved.

The dynamics of solving AP problems start by identifying what portions of the information pertain to the given formulas, such as the nth term and the sum formulas. The approach usually involves setting up the relevant equations based on the provided information and then solving these equations.

Consider the two key steps:

Setting up Equations

Beginning with understanding the given terms or sums and then translating them into algebraic expressions using the AP formulas. Proper identification of 'a', 'd', and 'n' is essential.

Solving the Equations

This step may involve manipulating the expressions, employing algebraic techniques such as substitution or elimination, and isolating the desired variable(s). It often helps to simplify the equation by consolidating like terms or factoring.

Prioritizing understanding the logic behind these steps over rote memorization of formulas leads to a more profound comprehension of the material, and equips students with the tools to tackle various arithmetic progression problems with confidence.