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.