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The task presents two savings options over a series of weeks. Here's how each option can be outlined and calculated:

### Option 1: Fixed Addition
- Each week, you add a fixed amount of 20 pesos to your savings.
- If you start saving on the 1st week of October:
  - **1st week**: 20 pesos
  - **2nd week**: 20 pesos + 20 pesos = 40 pesos
  - **3rd week**: 40 pesos + 20 pesos = 60 pesos
  - Continue this process through to the end of December.

### Option 2: Doubling the Amount
- Each week, you double the amount saved from the previous week.
- Starting with a base amount, let's say 20 pesos:
  - **1st week**: 20 pesos
  - **2nd week**: 20 pesos × 2 = 40 pesos
  - **3rd week**: 40 pesos × 2 = 80 pesos
  - Continue doubling each week until the end of December.

### Calculating Each Option
1. **Option 1 (Fixed Addition):**
   - Total savings over a period = 20 + 20 + 20 + ... (for each week)

2. **Option 2 (Doubling):**
   - This follows a geometric progression where each term is twice the previous one. The sum of a geometric series is given by:
     \[
     S_n = a \times \frac{r^n - 1}{r - 1}
     \]
   Where:
   - \( a \) = first term
   - \( r \) = common ratio (2 in this case)
   - \( n \) = number of terms (weeks in this case)

### Illustrating Savings on the Table
To fill in the table:
- Calculate the weekly savings for each option and add them to get the cumulative total.
- Sum up the weekly savings for each month (October, November, December) to get the final total.

Would you like a detailed step-by-step calculation for each week, or do you have specific questions about how to set up this problem?

### 5 Related Questions
1. How would the total savings differ if the fixed addition amount were 30 pesos instead of 20?
2. How does changing the initial amount for the doubling option affect the total savings?
3. What is the formula for calculating the sum of the savings for the fixed addition option?
4. How would the savings change if we applied a different doubling rate, like tripling instead?
5. How can this savings challenge be adapted to different time intervals, like bi-weekly or monthly?

### Tip
When calculating savings or investments with a regular increase or a doubling factor, recognizing whether it's an arithmetic or geometric sequence is key to finding the correct formula to use.

The task is to choose between two savings options over several weeks. The first option involves adding a fixed amount of 20 pesos each week. The second option involves doubling the amount saved each week. The goal is to determine which option yields more savings by the end of December.

This Pre-Calculus task involves deciding between two savings strategies: adding a fixed amount weekly or doubling the amount saved each week. Learn how to calculate and compare the total savings using arithmetic and geometric sequences. Suitable for Grades 10-12.

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