**The Most Difficult Question in Arithmetic Propagation (AP)**

Arithmetic Propagation (AP) is a fundamental concept in mathematics that deals with sequences of numbers where each term is obtained by adding a constant difference to the previous term. While AP problems can vary in difficulty, one question stands out as particularly challenging: finding the value of the nth term in an AP sequence when only a few terms are given.

**Problem Statement:**
Consider an AP sequence with the first term (a), common difference (d), and the nth term (an) unknown. However, we are given the values of two terms: the pth term (ap) and the qth term (aq). The task is to determine the value of the nth term (an) using this limited information.

**Solution Approach:**
To solve this problem, we need to employ the concept of arithmetic progression and utilize the given information effectively.

1. **Identify the Known Values:**
Determine the values provided in the problem:
- First term (a)
- Common difference (d)
- Terms whose values are known (ap and aq)

2. **Find the Positions of Known Terms:**
Calculate the positions of the known terms in the sequence. This can be done by dividing the difference between the terms (q - p) by the common difference (d) and adding 1.
- Position of ap: p + 1
- Position of aq: q + 1

3. **Derive the Equations:**
Using the known values and positions, form two equations:
- Equation 1: ap = a + (p + 1 - 1) * d
- Equation 2: aq = a + (q + 1 - 1) * d

4. **Eliminate 'a' from the Equations:**
Subtract Equation 1 from Equation 2 in order to eliminate the 'a' term:
- aq - ap = (q - p) * d

5. **Solve for 'd':**
Divide both sides of the equation by (q - p) to obtain the common difference 'd':
- d = (aq - ap) / (q - p)

6. **Substitute 'd' into the Equation:**
Replace the value of 'd' in either Equation 1 or Equation 2 to find 'a':
- a = ap - (p + 1 - 1) * d

7. **Find the nth Term:**
Finally, substitute the values of 'a' and 'd' into the equation for the nth term:
- an = a + (n + 1 - 1) * d

Applying this step-by-step approach enables us to determine the value of the nth term in an AP sequence when only two terms are given. It requires a thorough understanding of arithmetic progression and the ability to manipulate equations to solve for unknowns.

Remember, practice is key to mastering AP problems, so attempt various examples and reinforce your understanding of the underlying concepts.

There are some twisted questions in Arithmetic progressions.example:-Prove that nth term of An A.P cannot be n×n+1 or n^2+1.Justify your answer