To model this problem, we'll set up a formula that accounts for both the water added on odd days and the water removed on even days. The key variables are:

• Day $n$: Refers to the day number.
• Odd days: Water added increases by 2 liters starting from 50 liters on day 1.
• Even days: Water removed starts at 50 liters on day 2, reducing by 10% on each consecutive even day.

Let's define the following:

• $A(n)$: Amount of water added on the $n$-th odd day.
• $R(n)$: Amount of water removed on the $n$-th even day.
• $W_n$: Total amount of water in the tank after day $n$.

1. Water added on odd days:

• Day 1: $A(1) = 50$ liters.
• Day 3: $A(3) = 50 + 2 = 52$ liters.
• Day 5: $A(5) = 50 + 2 \times 2 = 54$ liters.
• General formula for water added on the $k$-th odd day:
  $A(n_{\text{odd}}) = 50 + 2 \times (k - 1)$ where $k$ is the index of the odd day (1st odd day, 2nd odd day, etc.).

2. Water removed on even days:

• Day 2: $R(2) = 50$ liters.
• Day 4: $R(4) = 50 \times (1 - 0.1) = 50 \times 0.9 = 45$ liters.
• Day 6: $R(6) = 45 \times (1 - 0.1) = 40.5$ liters.
• General formula for water removed on the $k$-th even day:
  $R(n_{\text{even}}) = 50 \times (0.9)^{k - 1}$ where $k$ is the index of the even day (2nd day, 4th day, etc.).

3. Total amount of water in the tank:

Starting from day 1 with 0 liters in the tank, the water added or removed on each day modifies the total amount. The total water in the tank after day $n$ is:

W_n = W_{n-1} + A(n) \quad \text{if n is odd} W_n = W_{n-1} - R(n) \quad \text{if n is even}

Where $W_0 = 0$ liters.

4. Overflow condition:

The tank has a capacity of 500 liters, so the tank will overflow when $W_n > 500$.

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To find when the tank overflows, you will need to iteratively apply the water added and removed formulas to compute the total water for each day, continuing until $W_n$ exceeds 500 liters.

Would you like me to compute the exact day when the tank overflows?

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5 Follow-Up Questions:

1. How does the amount of water added change over time?
2. How does the reduction in the removal rate on even days affect the tank's filling rate?
3. Can the tank overflow only due to water added, or do the removal rates slow this process down significantly?
4. How would the overflow day change if the initial amount of water added or removed was different?
5. How would the situation change if the tank’s capacity were smaller or larger?

Tip: When solving problems involving gradual change, always keep an eye on patterns in the increase or decrease of quantities, like the 2-liter increase or 10% reduction in this problem.