We are tasked with determining how many days the Department of Health's fund will last, given their budget and spending pattern.

Problem Breakdown:

• Budget: ₱3,312,000
• Spending pattern:
  • Day 1: ₱72,000
  • Day 2: ₱90,000
  • Day 3: ₱108,000
  • and so on, increasing by ₱18,000 daily.

This spending pattern represents an arithmetic sequence:

• First term ($a_1$) = ₱72,000
• Common difference ($d$) = ₱18,000

The goal is to find the number of days ($n$) before the budget is exhausted.

Step 1: Sum of the arithmetic sequence

The total spending after $n$ days can be expressed using the sum of an arithmetic sequence formula: $S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)$ Where:

• $S_n$ = total budget (₱3,312,000)
• $a_1 = 72,000$
• $d = 18,000$

Step 2: Plugging in the known values

We substitute these values into the equation: $3,312,000 = \frac{n}{2} \times (2(72,000) + (n - 1)(18,000))$ Simplifying: $3,312,000 = \frac{n}{2} \times (144,000 + 18,000n - 18,000)$ $3,312,000 = \frac{n}{2} \times (126,000 + 18,000n)$ $3,312,000 = n(63,000 + 9,000n)$

Step 3: Solve the quadratic equation

Expand and rearrange the equation: $9,000n^2 + 63,000n - 3,312,000 = 0$ Divide by 1,000 to simplify: $9n^2 + 63n - 3,312 = 0$ Solve this quadratic equation using the quadratic formula: $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where:

• $a = 9$
• $b = 63$
• $c = -3,312$

Let's calculate this next.The quadratic equation gives us two solutions: $n_1 = 16$ and $n_2 = -23$.

Since the number of days ($n$) cannot be negative, the valid solution is $n = 16$.

Conclusion:

The Department of Health's fund will last for 16 days.

Would you like more details or have any questions? Here are five related questions you could explore:

1. How would the solution change if the common difference was ₱20,000 instead of ₱18,000?
2. What if the initial budget was reduced to ₱2,500,000?
3. How can we derive the total spent after exactly 10 days?
4. Can we solve this problem using a different method, such as graphing?
5. What happens if the spending decreases instead of increases each day?

Tip: When dealing with arithmetic sequences, the sum formula is a powerful tool for solving problems related to cumulative growth or decline.