Problem 78 - Precalculus Mathematics for Calculus

Question

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

After 30 days, she has $10,737,418.24. She becomes a billionaire on day 37.

Step 1: Understanding the Pattern

The woman saves money in a pattern where each day she saves double the amount she saved the previous day. This pattern is geometric progression.

Step 2: Identify the Formula for Total Savings

The amount saved on the nth day can be expressed as \( 2^{n-1} \) cents. Therefore, the total amount saved over 30 days is the sum of a geometric series: \( S = 2^0 + 2^1 + 2^2 + ... + 2^{29} \).

Step 3: Sum of a Geometric Series

To find the sum of this series, use the formula for the sum of the first n terms of a geometric series: \( S_n = a \frac{r^n - 1}{r - 1} \), where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 1 \) cent and \( r = 2 \).

Step 4: Calculate Total Savings for 30 Days

Plugging into the formula:\[ S_{30} = 1 \frac{2^{30} - 1}{2 - 1} = 2^{30} - 1 \] cents. Calculate \( 2^{30} \) to find the total in cents.

Step 5: Convert Total to Dollars

Convert the total cents into dollars. Since \( 1 \) dollar is \( 100 \) cents, divide the total calculated cents by \( 100 \).

Step 6: Find the Day She Becomes a Billionaire

A billionaire has \( 1 \) billion dollars, i.e., \( 10^9 \) cents. We need to find \( n \) such that the sum of savings equals or exceeds \( 10^9 \) cents: \( 2^n - 1 \geq 10^9 \). Solve the inequality for \( n \).

Key Concepts

Geometric Series

The original exercise showcases a scenario where money is saved daily following a specific increasing pattern. This pattern is known as a **geometric progression**. In a geometric series, each term after the first is obtained by multiplying the previous term by a constant called the **common ratio**. In our exercise, the woman doubles her savings each day, which means the common ratio is 2.
A geometric series itself is essentially the sum of the terms of a geometric progression. For example, if we look at individual days, she saves 1 cent on day one, 2 cents on day two, 4 cents on day three, and this keeps going. These values are not arbitrary but follow a structured mathematical pattern, making it easier to calculate large runs of terms, like the entire month's 30 days.
Understanding geometric series helps simplify many patterns and calculations, especially those involving exponential growth like interest rates, population growth, and even binary systems in computing.

Sum of a Geometric Series

To find out how much the woman saved at the end of 30 days, we calculate the **sum of the geometric series**. This sum is computed using a specific formula:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, and \( r \) is the common ratio. For this exercise, \( a = 1 \) cent, \( r = 2 \), and \( n = 30 \). Plug these into the formula to find:
\[ S_{30} = 1 \frac{2^{30} - 1}{2 - 1} \]
This makes calculating large numbers feasible without having to manually add each day's savings. It is a great showcase of how the formula can effectively summarize what could be a tedious process by hand. By showcasing this, students can appreciate how mathematicians simplify tasks using known formulas and series.

Mathematical Pattern Analysis

Mathematical pattern analysis allows us to break down and understand complex sequences like the one in this exercise. Recognizing a pattern, like the one formed by doubling each day's savings, enables us to apply analytical methods and mathematical formulas to solve problems more easily.
Here are key components of using mathematical pattern analysis effectively:

• **Identify the Pattern:** Determine if each term relates in a specific way to previous ones. Here, we see that each day’s savings is double that of the last.
• **Choose the Right Formula:** By understanding the pattern as geometric, we can use the geometric series formula for sums.
• **Predict Future Terms or Sums:** With the pattern formula known, calculate any future term or desired total, revealing insights without exhaustive computation.

By dissecting the patterns in real-life problems, students can find solutions more efficiently, uncovering the math's predictive power in everyday contexts. Mathematical pattern analysis is not just about dealing with numbers; it's about recognizing systems that allow for calculation simplification and anticipation, whether in finance, science, or everyday decision-making.