Question: Activity C.1: Saving Up Smartly A Real-Life Application of Sequences SCENARIO: A student wants to build a personal savings fund. They commit to saving ₱50.00 in the first week and increase the amount they save by ₱15.00 each week. ❖ Week 1: ₱50.00 ❖ Week 2: ₱65.00 ❖ Week 3: ₱80.00 ❖ ... (This continues as an arithmetic sequence.)

1. **Problem Statement:** A student saves money weekly starting with $50$ in the first week and increases the saving by $15$ each subsequent week. This forms an arithmetic sequence. 2. **Formula for Arithmetic Sequence:** The $n$th term of an arithmetic sequence is given by: $$ a_n = a_1 + (n-1)d $$ where $a_1$ is the first term and $d$ is the common difference. 3. **Identify values:** - First term $a_1 = 50$ - Common difference $d = 15$ 4. **Example: Calculate the amount saved in Week 4:** $$ a_4 = 50 + (4-1) \times 15 = 50 + 3 \times 15 = 50 + 45 = 95 $$ 5. **Total savings after $n$ weeks:** The sum of the first $n$ terms of an arithmetic sequence is: $$ S_n = \frac{n}{2} (2a_1 + (n-1)d) $$ 6. **Example: Total saved after 4 weeks:** $$ S_4 = \frac{4}{2} (2 \times 50 + (4-1) \times 15) = 2 (100 + 45) = 2 \times 145 = 290 $$ 7. **Explanation:** Each week, the student saves $15$ more than the previous week, starting at $50$. Using the formulas, we can find the amount saved any week and the total saved over multiple weeks. **Final answers:** - Amount saved in Week 4 is $95$. - Total saved after 4 weeks is $290$.