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. **State the problem:** A student saves money weekly starting with $50$ in the first week and increases the amount saved by $15$ each subsequent week. This forms an arithmetic sequence.
2. **Formula for arithmetic sequence:** The $n$th term $a_n$ 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:** Here, $a_1 = 50$ and $d = 15$.
4. **Example calculation:** To find the amount saved in week $n$, substitute values:
$$a_n = 50 + (n-1)15$$
5. **Explanation:** This means each week the student saves $15$ more than the previous week, starting at $50$.
6. **Total savings after $n$ weeks:** The sum $S_n$ of the first $n$ terms of an arithmetic sequence is
$$S_n = \frac{n}{2} (2a_1 + (n-1)d)$$
7. **Example:** Total saved after $4$ weeks:
$$S_4 = \frac{4}{2} (2\times 50 + (4-1)15) = 2 (100 + 45) = 2 \times 145 = 290$$
8. **Interpretation:** After $4$ weeks, the student will have saved a total of $290$.
This approach helps plan savings over time using arithmetic sequences.