Problem 22 - Algebra for College Students

Question

Suppose that you save a dime the first day of a month, \(\$ 0.20\) the second day, and \(\$ 0.40\) the third day and that you continue to double your savings each day for 14 days. Find the total amount that you will save at the end of 14 days. \(\quad \$ 1638.30\)

Step-by-Step Solution

Answer

You will save a total of \(\$1638.30\) at the end of 14 days.

Step 1: Understand the pattern

Recognize that each day, the amount saved is doubled from the previous day. This sequence starts with \(0.10\) dollars on the first day.

Step 2: Identify the sequence

The savings form a geometric progression where the first term \(a = 0.10\) dollars and the common ratio \(r = 2\).

Step 3: Use the formula for the sum of a geometric series

The sum \(S_n\) of the first \(n\) terms of a geometric series is given by \(S_n = a \frac{r^n - 1}{r - 1}\). In this case, \(a = 0.10\), \(r = 2\), and \(n = 14\).

Step 4: Plug values into the formula

Substitute \(a = 0.10\), \(r = 2\), and \(n = 14\) into the formula: \(S_{14} = 0.10 \frac{2^{14} - 1}{2 - 1}\).

Step 5: Calculate \(2^{14}\)

Calculate \(2^{14} = 16384\).

Step 6: Calculate the sum

Use the result from Step 5 to find \(S_{14} = 0.10 \times (16384 - 1) = 0.10 \times 16383 = 1638.30\).

Step 7: Conclusion

The total amount saved after 14 days is \(\$1638.30\).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. This pattern creates a progression that can be found in various real-world problems, including financial applications and savings plans. In a geometric series, you begin with an initial term, and each subsequent term is the initial term multiplied by the common ratio raised to the appropriate power.
For example, in the saving money sequence described here, you start with saving $0.10 on the first day. Each day, the amount saved doubles, indicating the common ratio is 2. This constant doubling creates the sequence 0.10, 0.20, 0.40, and so on. Observing this pattern helps you realize that you are dealing with a geometric series.
Understanding this type of series is crucial for solving such exercises efficiently as it simplifies calculations and provides a systematic approach to derive totals.

Sum of Series Formula

The sum of a geometric series can be calculated using a simple but powerful formula. This formula serves as a tool to find the total of the first few terms in a geometric series without manually adding each term.
To grasp this, consider the formula for the sum of the first \(n\) terms of a geometric series: \[S_n = a \frac{r^n - 1}{r - 1}\]- \(a\) is the first term of the series.- \(r\) is the common ratio between consecutive terms.- \(n\) is the number of terms you want to sum.
For the saving money sequence in the exercise, substituting \(a = 0.10\), \(r = 2\), and \(n = 14\) into the formula gives:\[S_{14} = 0.10 \times \frac{2^{14} - 1}{1}\]Calculating this, first you compute \(2^{14}\) which is 16384, then:\[S_{14} = 0.10 \times (16384 - 1) = 0.10 \times 16383 = \$1638.30\]
This formula streamlines calculations in geometric series, making it perfect for efficiently solving tasks that follow such a pattern.

Saving Money Sequence

In real-life, finding creative ways to save money can be both challenging and rewarding. The saving money sequence presented in this exercise is not just a mathematical curiosity but also a clever way to demonstrate the power of exponential growth.
When you double your savings every day, as shown in the problem, it teaches a valuable lesson about how small amounts can grow rapidly over time. By day 14, starting from saving just \(0.10, you accumulate a total of \\)1638.30. This incredible growth showcases the effects of a geometric progression in practice.
Implementing sequences like these helps instill habits of regular saving, starting with a minimal initial amount and possibly increasing it systematically, like doubling in this case. This could be a fun and structured way to encourage consistent saving. Moreover, understanding how such sequences operate, and using tools like the sum of series formula, allows for more strategic financial planning and awareness of the long-term benefits of saving.