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Let's work through these Arithmetic Progression (AP) questions one by one.

### (1) If \(a = 5\), \(d = 3\), and \(a_n = 50\), find \(n\) and \(S_n\).

The general formula for the \(n\)-th term of an AP is:

\[
a_n = a + (n - 1) \cdot d
\]

Substituting the values:

\[
50 = 5 + (n - 1) \cdot 3
\]

Simplifying this equation:

\[
50 - 5 = (n - 1) \cdot 3
\]

\[
45 = (n - 1) \cdot 3
\]

\[
n - 1 = 15 \quad \Rightarrow \quad n = 16
\]

Now, using the formula for the sum of the first \(n\) terms of an AP:

\[
S_n = \frac{n}{2} \cdot (2a + (n-1)d)
\]

Substituting the known values:

\[
S_{16} = \frac{16}{2} \cdot (2 \cdot 5 + (16 - 1) \cdot 3)
\]

\[
S_{16} = 8 \cdot (10 + 45) = 8 \cdot 55 = 440
\]

So, \(n = 16\) and \(S_n = 440\).

---

### (2) If \(a_n = 4\), \(d = 2\), and \(S_n = -14\), find \(n\) and \(a\).

We use the two equations again.

1. For \(a_n\):

\[
a_n = a + (n - 1) \cdot d
\]

Substituting \(a_n = 4\) and \(d = 2\):

\[
4 = a + (n - 1) \cdot 2
\]

2. For \(S_n\):

\[
S_n = \frac{n}{2} \cdot (2a + (n-1) \cdot d)
\]

Substitute \(S_n = -14\) and \(d = 2\):

\[
-14 = \frac{n}{2} \cdot (2a + (n-1) \cdot 2)
\]

---

### (3) If \(a = 3\), \(n = 8\), and \(S_n = 192\), find \(d\).

The sum formula is:

\[
S_n = \frac{n}{2} \cdot (2a + (n-1) \cdot d)
\]

Substitute the known values:

\[
192 = \frac{8}{2} \cdot (2 \cdot 3 + (8-1) \cdot d)
\]

---

### (4) If \(a_n = 28\), \(S_n = 144\), and \(n = 9\), find \(a\).

1. Use \(a_n = a + (n - 1) \cdot d\).
2. Use the sum formula \(S_n = \frac{n}{2} \cdot (2a + (n-1) \cdot d)\).

---

Would you like detailed solutions for questions 2 to 4? Let me know if you have any questions.

Here are five related questions for further practice:
1. What is the formula for the sum of an infinite arithmetic series?
2. How would you find the common difference if only two terms are given?
3. How can you calculate the first term if you know the last term, number of terms, and sum?
4. What is the meaning of \(a_n\) in an arithmetic progression?
5. Can the sum of an arithmetic series be negative?

**Tip:** Always remember that in an AP, the difference between consecutive terms is constant.

Answer the following as per standard notations of an AP: (1) If a = 5, d = 3 and an = 50, find n and Sn. (2) If an = 4, d = 2 and Sn = -14, find n and a. (3) If a = 3, n = 8 and Sn = 192, find d. (4) If an = 28, Sn = 144 and n = 9, find a.

Explore detailed solutions to standard arithmetic progression problems, including finding terms, sums, and other key parameters of sequences. Suitable for high school students, this guide uses common AP formulas like nth term and sum of terms.

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