Problem 78 - Algebra and Trigonometry

Question

Geometric Savings Plan A very patient woman wishes to become a billionaire. She decides to follow a simple scheme: She puts aside 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on, doubling the number of cents each day. How much money will she have at the end of 30 days? How many days will it take this woman to realize her wish?

Step-by-Step Solution

Answer

At the end of 30 days, she will have $10,737,418.23. It will take 38 days to become a billionaire.

Step 1: Identify the Pattern

The woman saves by doubling the amount from the previous day. This forms a geometric sequence where the first term is 1 cent and each subsequent term is multiplied by 2. Thus, the sequence can be denoted as \( a_1 = 1 \), \( a_2 = 2 \), \( a_3 = 4 \), ..., \( a_n = 2^{(n-1)} \).

Step 2: Calculate Total Savings in 30 Days

The total savings after 30 days is the sum of the geometric sequence from day 1 to day 30. The formula for the sum of the first \( n \) terms of a geometric sequence is \( S_n = a_1 \frac{r^n - 1}{r - 1} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. Here, \( a_1 = 1 \), \( r = 2 \), and \( n = 30 \). So, \( S_{30} = 1 \times \frac{2^{30} - 1}{2 - 1} = 2^{30} - 1 \) cents.

Step 3: Convert Cents to Dollars

The result from the previous step is in cents. To convert it to dollars, divide by 100. Thus, the amount is \( \frac{2^{30} - 1}{100} \) dollars.

Step 4: Calculate Amount in Dollars

First, calculate the value of \( 2^{30} \). This can be computed as \( 1,073,741,824 \). Subtract 1 to get 1,073,741,823 cents. Then convert to dollars: \( \frac{1,073,741,823}{100} = 10,737,418.23 \) dollars.

Step 5: Determine Needed Days to Reach a Billion

To find out how many days it takes to reach at least one billion dollars (1,000,000,000 dollars or 100,000,000,000 cents), we need to find \( n \) such that \( 2^n - 1 \geq 100,000,000,000 \). Using the inequality, solve for \( n \): \( 2^n \geq 100,000,000,001 \). By taking logarithms, \( n \log(2) \geq \log(100,000,000,001) \) which gives \( n \geq \frac{\log(100,000,000,001)}{\log(2)} \approx 37.51 \).

Step 6: Round Up Days Calculation

Since the number of days must be a whole number and \( n \geq 37.51 \), round up to find that she needs 38 days to have at least one billion dollars.

Key Concepts

Geometric Savings Plan

The concept of a geometric savings plan is intriguing and can be likened to a practical application of the mathematical theory of geometric sequences. In this type of plan, the amount saved follows a specific pattern: each day's savings are a multiple of the previous day's amount. For the plan in our problem, the woman starts by saving 1 cent on the first day and then doubles that amount every following day. This effectively creates a geometric sequence of daily savings.

This approach demonstrates exponential growth in savings, as each increment is based on multiplication rather than addition. Such a savings strategy, although starting very modestly, can grow rapidly to significant amounts due to the nature of geometric progression.

Sum of Geometric Sequence

Understanding the sum of a geometric sequence is crucial to calculate the total amount saved in a given period. In our example, the woman's savings form a geometric sequence where each term is twice the previous. To find out how much she saves in 30 days, we use the formula:

• Sum of the first \( n \) terms: \( S_n = a_1 \frac{r^n - 1}{r - 1} \)

where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. For her savings:
- \( a_1 = 1 \) cent
- Common ratio \( r = 2 \)
- Number of terms \( n = 30 \)
By substituting these into the formula, we get \( S_{30} = 2^{30} - 1 \) cents saved after 30 days. Converting this to dollars gives a handy approximation of how much she accumulates by the end of the month.

Compound Interest

Though not directly evident in the given exercise, the underlying principles of a geometric savings plan echo the growth observed in compound interest scenarios. Compound interest functions by having an initial sum that earns interest, which then accumulates on both the original sum and the added interest from previous periods.

Geometrically doubling savings acts in a similar fashion. Each day's savings build upon all the previous ones, just as compound interest accumulates based on previous interest earnings. This exponential nature is what turns seemingly tiny beginnings into vast sums over repeated cycles.

Exponential Growth

The exercise vividly illustrates the power of exponential growth. By understanding this concept, one can appreciate how small, consistent increases can develop into large outcomes in less time than expected. Exponential growth occurs when a quantity grows at a consistent continuous rate. In this problem, the savings double every day, reflecting a perfect example of exponential doubling time.

• The time it takes for a quantity to double is known as the doubling period.
• This concept is seen in various fields such as population growth, finance, and technology improvements.

With each step, the woman's savings leap to double their previous size. It’s this element of exponential growth that allows her savings to reach 10,737,418.23 dollars in just 30 days and suggests that she would become a billionaire shortly thereafter.