I have a lesson to do every day. If I do not complete it that very day, it turns overdue. I cannot complete any other lessons after that overdue lesson until I complete the overdue. I have eight overdue lessons already, plus the lesson for each day, and I complete two overdue lessons each day to catch up, but the lesson for each day turns overdue.. Will I eventually catch up, and if so, how long would it take to catch up?

Answer 1

To determine if you will eventually catch up and how long it will take, we need to set up the situation with some variables.

1. You currently have 8 overdue lessons.
2. Every day, you also have 1 new lesson to complete.
3. You can complete 2 overdue lessons each day.

So, to summarize:

• On day 1, you start with 8 overdue lessons.
• Each day, you finish 2 of those overdue lessons, but you also get 1 new lesson which adds to your overdue count.

Now let's calculate the situation day by day:

• Day 1:

  • Start: 8 overdue
  • Complete: 2 overdue lessons
  • New lesson: +1 (turns overdue)
  • End of day: \(8 - 2 + 1 = 7\) overdue lessons remaining.

• Day 2:

  • Start: 7 overdue
  • Complete: 2 overdue lessons
  • New lesson: +1 (turns overdue)
  • End of day: \(7 - 2 + 1 = 6\) overdue lessons remaining.

• Day 3:

  • Start: 6 overdue
  • Complete: 2 overdue lessons
  • New lesson: +1 (turns overdue)
  • End of day: \(6 - 2 + 1 = 5\) overdue lessons remaining.

• Day 4:

  • Start: 5 overdue
  • Complete: 2 overdue lessons
  • New lesson: +1 (turns overdue)
  • End of day: \(5 - 2 + 1 = 4\) overdue lessons remaining.

• Day 5:

  • Start: 4 overdue
  • Complete: 2 overdue lessons
  • New lesson: +1 (turns overdue)
  • End of day: \(4 - 2 + 1 = 3\) overdue lessons remaining.

• Day 6:

  • Start: 3 overdue
  • Complete: 2 overdue lessons
  • New lesson: +1 (turns overdue)
  • End of day: \(3 - 2 + 1 = 2\) overdue lessons remaining.

• Day 7:

  • Start: 2 overdue
  • Complete: 2 overdue lessons
  • New lesson: +1 (turns overdue)
  • End of day: \(2 - 2 + 1 = 1\) overdue lesson remaining.

• Day 8:

  • Start: 1 overdue
  • Complete: 1 overdue lesson
  • New lesson: +1 (turns overdue)
  • End of day: \(1 - 1 + 1 = 1\) overdue lesson remaining.

• Day 9:

  • Start: 1 overdue
  • Complete: 1 overdue lesson
  • New lesson: +1 (turns overdue)
  • End of day: \(1 - 1 + 1 = 1\) overdue lesson remaining.

• Day 10:

  • Start: 1 overdue
  • Complete: 1 overdue lesson
  • New lesson: +1 (turns overdue)
  • End of day: \(1 - 1 + 1 = 1\) overdue lesson remaining.

From Day 8 onward, it becomes clear that you will never reduce the overdue count to zero because you only complete 1 overdue lesson each day while adding 1 new lesson to the overdue count.

Thus, with the current system, you will never catch up as long as you maintain the same rate of completing lessons and receiving new lessons.