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Let's break down and solve both problems from the image.

### Question 12:
An arithmetic progression (AP) has 20 terms. Its first term is 10, and the sum of all its terms is 580. Find the value of the last term and the common difference.

Given:
- \( a = 10 \) (first term)
- \( n = 20 \) (number of terms)
- \( S_n = 580 \) (sum of the terms)

We use the formula for the sum of an arithmetic progression:
\[
S_n = \frac{n}{2} \times (a + l)
\]
where \( l \) is the last term. Plugging in the values:
\[
580 = \frac{20}{2} \times (10 + l)
\]
\[
580 = 10 \times (10 + l)
\]
\[
58 = 10 + l
\]
\[
l = 48
\]

The last term \( l = 48 \).

Next, to find the common difference \( d \), we use the formula for the \( n \)-th term of an AP:
\[
l = a + (n-1) \cdot d
\]
\[
48 = 10 + (20-1) \cdot d
\]
\[
48 = 10 + 19 \cdot d
\]
\[
38 = 19 \cdot d
\]
\[
d = 2
\]

Thus, the common difference \( d = 2 \) and the last term is 48.

---

### Question 13:
The ninth term of an AP is 28, and the sum of the first 15 terms is 330. Find the sum of the first 10 terms.

Given:
- \( a_9 = 28 \) (ninth term)
- \( S_{15} = 330 \) (sum of the first 15 terms)

1. **Finding the common difference:**

The formula for the \( n \)-th term of an AP is:
\[
a_n = a + (n-1) \cdot d
\]
For the 9th term:
\[
a_9 = a + 8 \cdot d = 28
\]
So, 
\[
a + 8d = 28 \quad \text{(Equation 1)}
\]

2. **Sum of 15 terms:**

Using the sum formula:
\[
S_n = \frac{n}{2} \times (2a + (n-1) \cdot d)
\]
For \( n = 15 \):
\[
330 = \frac{15}{2} \times (2a + 14d)
\]
Simplifying:
\[
330 = 7.5 \times (2a + 14d)
\]
\[
44 = 2a + 14d \quad \text{(Equation 2)}
\]

Now, solving the system of equations:
From Equation 2: 
\[
2a + 14d = 44
\]
\[
a + 7d = 22 \quad \text{(Equation 3)}
\]

Subtract Equation 1 from Equation 3:
\[
(a + 7d) - (a + 8d) = 22 - 28
\]
\[
-d = -6 \quad \Rightarrow \quad d = 6
\]

Substituting \( d = 6 \) into Equation 1:
\[
a + 8 \times 6 = 28
\]
\[
a + 48 = 28
\]
\[
a = -20
\]

3. **Sum of the first 10 terms:**

Now that we have \( a = -20 \) and \( d = 6 \), we can find the sum of the first 10 terms:
\[
S_{10} = \frac{10}{2} \times (2a + (10-1) \cdot d)
\]
\[
S_{10} = 5 \times (2(-20) + 9 \cdot 6)
\]
\[
S_{10} = 5 \times (-40 + 54) = 5 \times 14 = 70
\]

Thus, the sum of the first 10 terms is 70.

---

Would you like more details or further clarifications on any step? 

Here are some related questions:
1. How do we derive the formula for the sum of an arithmetic progression?
2. What are some other methods to find the common difference in an AP?
3. How does the number of terms affect the sum in an arithmetic series?
4. How can we calculate the sum of the first few terms if the last term is known?
5. How can we use arithmetic progressions in real-life applications?

**Tip**: Always check the number of terms before applying the sum formula, as it changes the calculations significantly!

12. An AP has 20 terms. Its first term is 10 and the sum of all its terms is 580. Calculate the value of the last term and of the common difference.
13. The ninth term of an AP is 28 and the sum of its first 15 terms is 330. Find the sum of the first 10 terms.

Solve problems involving arithmetic progression (AP) with 20 terms, including finding the last term and common difference. Learn how to calculate the sum of the first 10 terms of an AP when the ninth term and sum of the first 15 terms are given. Suitable for Grades 10-12 students.

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