https://file.aiquickdraw.com/mathbot/file/1726623131268s8gxpk0i.jpg

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

### Problem 1:
**Diketahui deret aritmetika 60 + 50 + 40 + ... Tentukan jumlah 15 suku pertamanya!**

This is an arithmetic sequence where:
- First term (\(a\)) = 60
- Common difference (\(d\)) = 50 - 60 = -10

We are asked to find the sum of the first 15 terms.

The formula for the sum of the first \(n\) terms of an arithmetic sequence is:
\[
S_n = \frac{n}{2} \times \left( 2a + (n-1) \times d \right)
\]

Substitute the values \(n = 15\), \(a = 60\), and \(d = -10\) into the formula:
\[
S_{15} = \frac{15}{2} \times \left( 2 \times 60 + (15-1) \times (-10) \right)
\]

Let's calculate this.

### Problem 2:
**Dari sebuah deret aritmetika diketahui suku ketiga sama dengan 9, sedangkan jumlah suku kelima dan ketujuh sama dengan 36. Maka tentukan jumlah 10 suku pertamanya!**

Given:
- The third term (\(a_3\)) is 9.
- The sum of the fifth and seventh terms is 36.

We need to find the sum of the first 10 terms of the arithmetic sequence.

Using the formula for the \(n\)-th term of an arithmetic sequence:
\[
a_n = a + (n-1) \cdot d
\]
For the third term (\(a_3\)):
\[
a + 2d = 9 \quad \text{(1)}
\]

For the sum of the fifth and seventh terms:
\[
(a + 4d) + (a + 6d) = 36
\]
Simplify this equation:
\[
2a + 10d = 36 \quad \text{(2)}
\]

Now we solve the system of equations (1) and (2) to find \(a\) and \(d\), and then calculate the sum of the first 10 terms.

I'll proceed with solving both problems step by step.Here are the solutions for both problems:

### Problem 1:
The sum of the first 15 terms of the arithmetic sequence \(60 + 50 + 40 + \dots\) is:
\[
S_{15} = -150
\]
This indicates that after 15 terms, the sum of the sequence decreases below zero due to the negative common difference.

### Problem 2:
By solving the system of equations, we found:
- First term \(a = 13\)
- Common difference \(d = -2\)

Using these values, the sum of the first 10 terms of the arithmetic sequence is:
\[
S_{10} = 165
\]

Let me know if you'd like further clarification or details!

#### Related questions:
1. What is the formula for the \(n\)-th term in an arithmetic sequence?
2. How do you determine the sum of an arithmetic series when given two non-consecutive terms?
3. How does the common difference affect the behavior of an arithmetic sequence?
4. Can an arithmetic sequence have a negative sum, and under what conditions?
5. What is the relationship between arithmetic and geometric sequences?

#### Tip:
In arithmetic sequences, understanding the first term and common difference allows you to predict the behavior of the sequence.

1. Diketahui deret aritmetika 60 + 50 + 40 + ... Tentukan jumlah 15 suku pertamanya!
2. Dari sebuah deret aritmetika diketahui suku ketiga sama dengan 9, sedangkan jumlah suku kelima dan ketujuh sama dengan 36. Maka tentukan jumlah 10 suku pertamanya!

Learn how to solve arithmetic sequence problems such as finding the sum of the first 15 terms and determining the sum of the first 10 terms, with step-by-step solutions. The questions involve key concepts like summation of series and the nth term formula. Suitable for students in Grades 10-12.

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