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**13. The \( n \)-th term of an AP is \( 1 + 4n \). Find, in terms of \( n \), the sum of the first \( n \) terms of the AP.**

We are given the nth term of an arithmetic progression (AP) as \( a_n = 1 + 4n \), and we need to find the sum of the first \( n \) terms, denoted as \( S_n \).

### Solution Steps:

1. **Identify the first term (a) and the common difference (d)**:
   - The first term \( a_1 \) is the value of the AP when \( n = 1 \):
     \[
     a_1 = 1 + 4(1) = 1 + 4 = 5
     \]
   - The common difference \( d \) is the difference between consecutive terms:
     \[
     d = a_2 - a_1 = \left(1 + 4(2)\right) - (1 + 4(1)) = 9 - 5 = 4
     \]
     So, \( d = 4 \).

2. **Sum of the first \( n \) terms of an AP**:
   The formula for the sum of the first \( n \) terms \( S_n \) of an arithmetic progression is:
   \[
   S_n = \frac{n}{2} \left(2a + (n - 1)d \right)
   \]
   Substituting the values of \( a = 5 \) and \( d = 4 \):
   \[
   S_n = \frac{n}{2} \left(2(5) + (n - 1) \cdot 4\right)
   \]
   Simplifying:
   \[
   S_n = \frac{n}{2} \left(10 + 4(n - 1)\right)
   \]
   \[
   S_n = \frac{n}{2} \left(10 + 4n - 4\right)
   \]
   \[
   S_n = \frac{n}{2} \left(4n + 6\right)
   \]
   \[
   S_n = \frac{n}{2} \cdot 2(2n + 3)
   \]
   \[
   S_n = n(2n + 3)
   \]

Thus, the sum of the first \( n \) terms of the arithmetic progression is:

\[
S_n = n(2n + 3)
\]

Let me know if you'd like further clarifications! Here are some related questions:

1. What is the general formula for the sum of an arithmetic progression?
2. How do we derive the common difference in an AP from consecutive terms?
3. What is the formula for the nth term of an arithmetic progression?
4. How does the common difference affect the sum of terms in an AP?
5. Can the nth term of an AP be a non-linear expression in real scenarios?

**Tip:** Always ensure you first identify the first term and the common difference when dealing with arithmetic progressions.

The nth term of an AP is 1 + 4n. Find, in terms of n, the sum of the first n terms of the AP.

Explore how to find the sum of the first n terms of an arithmetic progression (AP) where the nth term is 1 + 4n. Understand key concepts such as the nth term, common difference, and the formula for the sum of an AP. Suitable for students in Grades 10-12.

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