Question
A toy model manufacturing company is going to start work on a new model boat design. The company plans to produce 62 units in its first week and then increase production by 75 units each week for all future weeks. The total number of manufactured model boats can be modeled with the arithmetic sequence 62, 137, 212, 287, . . . . Following this sequence, how many model boats would be manufactured in the 12th week?(1 point)
The company could manufacture model boats in the 12th week.
The company could manufacture model boats in the 12th week.
Answers
GPT-4o mini
In the given arithmetic sequence, we start with \( a_1 = 62 \) (the number of boats manufactured in the first week) and the common difference \( d = 75 \) (the increase in production each week).
The formula for the \( n \)-th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1) \cdot d
\]
To find the number of model boats manufactured in the 12th week (\( n = 12 \)), we plug in the values:
\[
a_{12} = 62 + (12 - 1) \cdot 75
\]
Calculating this step-by-step:
1. Compute \( 12 - 1 = 11 \)
2. Then, calculate \( 11 \cdot 75 = 825 \)
3. Finally, add this to 62:
\[
a_{12} = 62 + 825 = 887
\]
Thus, the company would manufacture **887 model boats** in the 12th week.
The formula for the \( n \)-th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1) \cdot d
\]
To find the number of model boats manufactured in the 12th week (\( n = 12 \)), we plug in the values:
\[
a_{12} = 62 + (12 - 1) \cdot 75
\]
Calculating this step-by-step:
1. Compute \( 12 - 1 = 11 \)
2. Then, calculate \( 11 \cdot 75 = 825 \)
3. Finally, add this to 62:
\[
a_{12} = 62 + 825 = 887
\]
Thus, the company would manufacture **887 model boats** in the 12th week.