Question

1. a concert venue has 20 seats in the first row and an 5 additional seats in each successive row. if there are 35 rows of seats, how many people can the venue seat?

• 2,100 people
• 190 people
• 7,350 people
• 3,675 people

Explanation:

🆕 New Concept Discovered: Arithmetic Series
Adding up terms of an arithmetic sequence

Step 1: Identify the sequence parameters

The number of seats in each row forms an arithmetic sequence where:

• First term \( a_1 = 20 \) (seats in the first row)
• Common difference \( d = 5 \) (additional seats in each successive row)
• Number of terms \( n = 35 \) (total number of rows)

Step 2: Find the number of seats in the last row

To find the number of seats in the 35th row (\( a_{35} \)), we use the arithmetic sequence formula:

$$ a_n = a_1 + (n - 1)d $$

Substitute the known values:

$$ a_{35} = 20 + (35 - 1) \times 5 $$

$$ a_{35} = 20 + 34 \times 5 $$

$$ a_{35} = 20 + 170 = 190 $$

Step 3: Calculate the total sum of seats

To find the total capacity, we sum the arithmetic series using the sum formula:

$$ S_n = \frac{n}{2} (a_1 + a_n) $$

Substitute the values:

$$ S_{35} = \frac{35}{2} (20 + 190) $$

$$ S_{35} = \frac{35}{2} (210) $$

$$ S_{35} = 35 \times 105 = 3,675 $$

Answer:

3,675 people