After an injury, your fitness instructors tell you to return to your jogging program slowly. They suggest jogging for 5 minutes for the first week. Each week thereafter, they tell you to increase that time by 3 minutes per week. Let a1 = 5 minutes, number of suggest jogging time for week 1; a2 = 8 mins, number of suggest jogging time for week 2; etc. How many minutes should you be jogging after two months (8 weeks)?

The problem describes an arithmetic sequence where the initial jogging time is 5 minutes, and the time increases by 3 minutes each week.

Let's break down the problem:

1. The first term of the sequence, \( a_1 = 5 \) minutes.
2. The common difference, \( d = 3 \) minutes.
3. The number of weeks (terms) is \( n = 8 \).

The general formula for the \( n \)-th term of an arithmetic sequence is:

\[
a_n = a_1 + (n - 1) \times d
\]

Substituting the given values:

\[
a_8 = 5 + (8 - 1) \times 3
\]

\[
a_8 = 5 + 7 \times 3
\]

\[
a_8 = 5 + 21 = 26 \text{ minutes}
\]

So, after 8 weeks, you should be jogging for **26 minutes**.

Would you like a detailed explanation of any step, or do you have any questions?

Here are 5 related questions to consider:
1. How would the sequence change if the weekly increase was 4 minutes instead of 3?
2. What would be the total jogging time over the 8 weeks?
3. How many weeks would it take to reach a jogging time of 50 minutes?
4. What if you start with 10 minutes instead of 5 minutes—how would the sequence change?
5. Can you express the total jogging time after \( n \) weeks using a formula?

**Tip:** When dealing with sequences, recognizing whether it's arithmetic or geometric helps in applying the correct formulas quickly.

Explore how to calculate jogging time progression using an arithmetic sequence example. Learn about arithmetic sequences, their formulas, and apply them to determine jogging times over weeks. Suitable for Grades 6-8.

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