https://file.aiquickdraw.com/mathbot/file/1726069260615ddhmt9qn.jpg

Let's solve the problems one by one from the image you uploaded.

### Question 21:
We are asked to find the sum of all multiples of 3 from 60 to 120. The multiples of 3 in this range are:

\[
60, 63, 66, 69, 72, \dots, 120
\]

This sequence is an arithmetic sequence where the first term \(a_1 = 60\), the common difference \(d = 3\), and the last term \(a_n = 120\).

1. **Number of terms (n):**

The formula for the \(n\)-th term of an arithmetic sequence is:

\[
a_n = a_1 + (n-1) \cdot d
\]

Setting \(a_n = 120\), we solve for \(n\):

\[
120 = 60 + (n-1) \cdot 3
\]
\[
120 - 60 = (n-1) \cdot 3
\]
\[
60 = (n-1) \cdot 3
\]
\[
n-1 = 20 \quad \Rightarrow \quad n = 21
\]

There are 21 terms in total.

2. **Sum of the arithmetic sequence:**

The formula for the sum \(S_n\) of an arithmetic sequence is:

\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]

Substituting the values:

\[
S_{21} = \frac{21}{2} \cdot (60 + 120) = \frac{21}{2} \cdot 180 = 21 \cdot 90 = 1890
\]

Thus, the sum of all multiples of 3 from 60 to 120 is **1,890**. Therefore, the correct answer is **D. 1,890**.

---

### Question 22:
We are asked to find the total number of chairs in an auditorium where:
- The first row has 11 chairs.
- Each subsequent row has 2 more chairs than the row before.
- There are 8 rows in total.

This describes an arithmetic sequence where:
- First term \(a_1 = 11\),
- Common difference \(d = 2\),
- Number of terms \(n = 8\).

1. **Number of chairs in each row:**

The number of chairs in each row follows this sequence:
\[
a_1 = 11, \quad a_2 = 13, \quad a_3 = 15, \dots, \quad a_8
\]

2. **Sum of the sequence:**

The sum of an arithmetic sequence is given by:

\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]

First, we find \(a_8\) using the formula for the \(n\)-th term of an arithmetic sequence:

\[
a_n = a_1 + (n-1) \cdot d
\]
\[
a_8 = 11 + (8-1) \cdot 2 = 11 + 14 = 25
\]

Now, the sum of the sequence is:

\[
S_8 = \frac{8}{2} \cdot (11 + 25) = 4 \cdot 36 = 144
\]

Thus, the total number of chairs is **144**. The correct answer is **A. 144**.

---

### Question 23:
We are told that a rope is cut into 6 pieces that form a geometric sequence. The shortest piece is 1 dm, and the longest piece is 243 dm. We need to find the total length of the rope.

Let the terms of the geometric sequence be \(a, ar, ar^2, ar^3, ar^4, ar^5\), where:
- \(a = 1\) dm is the shortest piece,
- \(ar^5 = 243\) dm is the longest piece.

1. **Find the common ratio (r):**

We know:

\[
ar^5 = 243 \quad \Rightarrow \quad 1 \cdot r^5 = 243 \quad \Rightarrow \quad r^5 = 243
\]

Taking the fifth root of both sides:

\[
r = \sqrt[5]{243} = 3
\]

2. **Sum of the geometric sequence:**

The sum \(S_n\) of the first \(n\) terms of a geometric sequence is given by:

\[
S_n = a \cdot \frac{r^n - 1}{r - 1}
\]

Substituting the values \(a = 1\), \(r = 3\), and \(n = 6\):

\[
S_6 = 1 \cdot \frac{3^6 - 1}{3 - 1} = \frac{729 - 1}{2} = \frac{728}{2} = 364
\]

Thus, the total length of the rope is **364 dm**. The correct answer is **D. 364 dm**.

---

Do you have any questions or would you like further clarification on any of these solutions?

Here are 5 related questions:
1. How do you derive the sum formula for an arithmetic series?
2. What is the general form of a geometric progression?
3. How do you find the sum of the first \(n\) terms in a geometric sequence?
4. What are the applications of arithmetic and geometric sequences in real life?
5. How would the sum formula change if the rope in question 23 had 7 pieces instead of 6?

**Tip:** When solving sequences, always identify whether it’s arithmetic or geometric first before applying formulas.

21. Jumlah semua bilangan kelipatan 3 dari 60 sampai 120 adalah ... 
22. Dalam suatu aula sekolah terdapat 11 kursi pada barisan pertama dan setiap baris berikutnya memuat dua kursi lebih banyak dari kursi di barisan depannya. Jika terdapat 8 baris kursi, maka banyak kursi dalam aula adalah ... 
23. Suatu tali dipotong menjadi 6 bagian yang membentuk deret geometri. Jika potongan terpendek adalah 1 dm dan potongan terpanjang adalah 243 dm, maka panjang tali semula adalah ...

This page solves three math problems involving arithmetic and geometric sequences. It covers how to find the sum of multiples of 3 from 60 to 120, calculate the number of chairs in an auditorium following an arithmetic sequence, and determine the total length of a rope cut into pieces forming a geometric sequence.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation