The Question
Given the arithmetic series: 5 + 7 + 9 + … + 93:
The series represents the number of kilometres that an athlete ran each week in preparation for an ultramarathon. The athlete ran 93 km in the last week of the training programme. How long, in weeks, was the training programme?
Details
💡 Hint
You’ve found the formula for T_n. Now, if the last term is 93, what does that tell you about ‘n’?
📝 Solution Steps
- Recall the general term T_n = 2n + 3 from Question 2.1.1.
- Set the general term T_n equal to the given value of 93 km.
- Solve the resulting linear equation for ‘n’ to find the number of weeks.
📚 Explanation
Building on the previous question, you have already determined the general term (T_n) for the arithmetic series. The problem states that the athlete ran 93 km in the *last* week. This means that 93 is the value of the nth term (T_n). To find out how long the training program was (i.e., the number of weeks, ‘n’), you need to set your derived T_n expression equal to 93 and solve the resulting linear equation for ‘n’.
✅ Answer
45 weeks
⚠️ Common Mistakes
- Algebraic errors when solving the linear equation for ‘n’.
- Misinterpreting ‘last week’ as the sum of the series instead of the nth term.
📐 Formulas Required
- T_n = a + (n-1)d
