Cinema 2 has 20 seats in the first row and 15 rows in all. Each successive row contains one additional seat. How many seats are in the theater?"

Let's solve this problem where Cinema 2 has:

- **20 seats in the first row**.
- **15 rows in total**.
- Each successive row contains **one additional seat**.

### Step-by-Step Solution:

1. **First term of the sequence**:  
   - \( a_1 = 20 \) (This is the number of seats in the first row).

2. **Number of rows (terms)**:  
   - \( n = 15 \).

3. **Common difference**:  
   - \( d = 1 \) (Each row has one more seat than the previous one).

### Sequence of Seats:
The number of seats in each row forms an arithmetic sequence:
- First row: 20 seats
- Second row: 21 seats
- Third row: 22 seats
- ...
- 15th row: \( 20 + (15 - 1) = 20 + 14 = 34 \) seats

### Sum of Seats:
To find the total number of seats, sum the sequence using the sum formula for an arithmetic sequence:
\[
S_n = \frac{n}{2} \times (2a_1 + (n-1)d)
\]
Substituting the known values:
\[
S_{15} = \frac{15}{2} \times (2 \times 20 + (15-1) \times 1) 
\]
\[
S_{15} = \frac{15}{2} \times (40 + 14) = \frac{15}{2} \times 54 = 7.5 \times 54 = 405
\]

So, the total number of seats in the theater is **405 seats**.

---

Would you like further explanation or details on this problem?

Here are some related questions:

1. What would be the total number of seats if there were 20 rows instead of 15?
2. How would the number of seats change if each successive row added 2 seats instead of 1?
3. Can you calculate the total number of seats if the sequence started with 25 seats in the first row?
4. What is the average number of seats per row in this theater?
5. How do you determine the total number of seats if the first row had 30 seats and each successive row reduced by 1 seat?
6. How would you calculate the total if the rows increased by a different common difference each time?
7. What is the difference between an arithmetic sequence and a geometric sequence?
8. How would the total number of seats change if only every second row added one seat?

**Tip:** When dealing with arithmetic sequences, always identify the first term, common difference, and the number of terms to easily solve for the sum or specific terms in the sequence.

Learn how to calculate the total number of seats in Cinema 2 using arithmetic sequences. Each row has a progressively increasing number of seats, starting with 20 seats in the first row and a total of 15 rows. Get a step-by-step solution and understand the sum formula for arithmetic sequences. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation