Answer :
To solve this problem, we need to determine the general term of the arithmetic sequence that represents Will's jogging time in minutes each week and find out how many weeks it takes for him to reach a jogging time of one hour (60 minutes).
1. Identify the Initial Conditions and Pattern:
- Will starts jogging 12 minutes on the first week.
- He increases his jogging time by 6 minutes each subsequent week.
2. Write the General Term:
- An arithmetic sequence has a general form [tex]\( a_n = a_1 + (n-1) \cdot d \)[/tex], where [tex]\( a_1 \)[/tex] is the initial term and [tex]\( d \)[/tex] is the common difference.
- Here, [tex]\( a_1 = 12 \)[/tex] minutes and [tex]\( d = 6 \)[/tex] minutes per week.
- Substitute these values into the general form:
[tex]\[
a_n = 12 + (n-1) \cdot 6
\][/tex]
- Simplify the equation:
[tex]\[
a_n = 12 + 6n - 6 = 6n + 6
\][/tex]
3. Determine the Week for 60 Minutes:
- Set the expression [tex]\( a_n \)[/tex] equal to 60 (minutes):
[tex]\[
6n + 6 = 60
\][/tex]
- Solve for [tex]\( n \)[/tex]:
[tex]\[
6n = 60 - 6
\][/tex]
[tex]\[
6n = 54
\][/tex]
[tex]\[
n = \frac{54}{6} = 9
\][/tex]
Thus, it takes 9 weeks for Will to reach a jogging time of one hour. Therefore, the correct choice based on this analysis is:
C. [tex]\( a_n = 6n + 6; 9 \)[/tex] weeks.
1. Identify the Initial Conditions and Pattern:
- Will starts jogging 12 minutes on the first week.
- He increases his jogging time by 6 minutes each subsequent week.
2. Write the General Term:
- An arithmetic sequence has a general form [tex]\( a_n = a_1 + (n-1) \cdot d \)[/tex], where [tex]\( a_1 \)[/tex] is the initial term and [tex]\( d \)[/tex] is the common difference.
- Here, [tex]\( a_1 = 12 \)[/tex] minutes and [tex]\( d = 6 \)[/tex] minutes per week.
- Substitute these values into the general form:
[tex]\[
a_n = 12 + (n-1) \cdot 6
\][/tex]
- Simplify the equation:
[tex]\[
a_n = 12 + 6n - 6 = 6n + 6
\][/tex]
3. Determine the Week for 60 Minutes:
- Set the expression [tex]\( a_n \)[/tex] equal to 60 (minutes):
[tex]\[
6n + 6 = 60
\][/tex]
- Solve for [tex]\( n \)[/tex]:
[tex]\[
6n = 60 - 6
\][/tex]
[tex]\[
6n = 54
\][/tex]
[tex]\[
n = \frac{54}{6} = 9
\][/tex]
Thus, it takes 9 weeks for Will to reach a jogging time of one hour. Therefore, the correct choice based on this analysis is:
C. [tex]\( a_n = 6n + 6; 9 \)[/tex] weeks.
