Question

1. after knee surgery, your trainer tells you to return to your jogging program slowly. she suggests jogging for 12 minutes each day for the first week. each week after that she suggests you increase that time by 6 minutes. how many weeks will it be before you are jogging one hour per day? type a response

Explanation:

Step1: Define variables and formula

Let \( n \) be the number of weeks after the first week. The time jogged each day forms an arithmetic sequence with first term \( a_1 = 12 \) minutes (first week), common difference \( d = 6 \) minutes, and we want to find when the term \( a_{n + 1}=60 \) minutes (1 hour). The formula for the \( k \)-th term of an arithmetic sequence is \( a_k=a_1+(k - 1)d \). Here, \( k=n + 1 \), so \( 60=12+(n + 1-1)\times6 \).

Step2: Simplify and solve for n

Simplify the equation: \( 60=12 + 6n \). Subtract 12 from both sides: \( 6n=60 - 12=48 \). Then divide by 6: \( n=\frac{48}{6}=8 \). But we need to include the first week, so total weeks \( =n + 1=8 + 1 = 9 \). Wait, wait, let's check again. Wait, the first week is week 1 with 12 minutes. Then week 2: 18, week 3:24,..., let's use the formula correctly. Let \( t \) be the time in minutes, \( w \) be the number of weeks. The formula is \( t = 12+6(w - 1) \), since for \( w = 1 \), \( t = 12 \), for \( w = 2 \), \( t = 18 \), etc. We want \( t = 60 \). So:

Step3: Solve \( 12+6(w - 1)=60 \)

Subtract 12: \( 6(w - 1)=48 \). Divide by 6: \( w - 1 = 8 \). Add 1: \( w = 9 \).

Answer:

9