Problem 65 - Algebra 1: Concepts and Skills

Question

Someone offers to double the amount of money you have every day for 1 month (30 days). You have 1 penny. At the end of the first day, you will have \(2 \cdot 1=2\) pennies. On the second day, you will have \(2 \cdot 2=4\) pennies. On the third day, you will have \(2 \cdot 4=8\) pennies. Write each of these equations using only powers of 2

Step-by-Step Solution

Answer

In this exponential growth scenario, on the nth day, the amount is \(2^n\) pennies.

Step 1: Recognizing the Pattern

Recognize the multiplication pattern. On the first day, you have \(2^1=2\) pennies. On the second day, you have \(2^2=4\) pennies, and on the third day, you have \(2^3=8\) pennies. Notice that the exponent corresponds to the day number.

Step 2: Applying the Pattern to Further Days

Continue applying the same pattern. The teacher will continue following this pattern until the end of the month. On the 30th day, you have \(2^{30}\) pennies.

Step 3: Understand the Exponential Growth Formula

Note how this relates to the formula for exponential growth, which is \(a=b \cdot c^{d}\), where 'a' is the final amount, 'b' is the initial amount, 'c' is the growth factor (2 in this case), and 'd' is the number of periods, in this case, 30 days. In this scenario, initially, there is 1 penny, each day the money is doubled hence the base is 2, and the increase happens over a duration of 30 days.

Key Concepts

Powers of 2

In everyday life, the idea of doubling might seem straightforward, but when it comes to mathematics, it's often expressed in terms of powers. Specifically, the phrase "powers of 2" emerges frequently. The power notation is a way of multiplying a number by itself a particular number of times. For example, when you have an expression like \(2^3\), this means you multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).

This concept of powers of 2 is especially useful in representing exponential growth patterns, which you might recognize in various scenarios, such as the growth of bacteria, computer science algorithms, or financial investments. Specifically, in the exercise above, each day is represented by a power of 2, with the exponent indicating the number of days. This leads to a quick calculation of large numbers, like day 30's \(2^{30}\), without having to perform repeated multiplications manually. Powers of 2 thus provide an elegant mathematical shorthand for exponential growth scenarios.

Doubling Pattern

The doubling pattern is a specific type of exponential growth that is commonly seen in real-world scenarios. Imagine starting with a small amount and simply doubling it at regular intervals. That's exactly what occurs in the exercise with the pennies. You start with 1 penny, double it each day, and the amount grows very rapidly.

This kind of pattern can be observed in nature as well, such as the spread of certain organisms like bacteria or even in some viral online trends. The key characteristic is how quickly the quantities grow. When something doubles constantly over equal intervals, it means that the increase compounds quickly. Starting from an initial amount, for example, one penny, if doubled each day, soon sees a rapid escalation in wealth through this powerful doubling mechanism. This is why understanding the doubling pattern is crucial for predictions and planning in many disciplines.

Exponential Function

Exponential functions model situations where quantities grow at a rate proportional to their current value, leading to rapid increases over time. The general form of an exponential function is \(a = b \cdot c^d\), where 'a' is the final value, 'b' is the starting amount, 'c' is the growth rate, and 'd' is the number of periods or steps.

In the context of the penny exercise, the exponential function helps illustrate how the initial 1 penny grows over 30 days with a consistent doubling (the growth rate being 2 each day). The equation \(a = 1 \cdot 2^{30}\) outlines how the 1 penny grows into a large sum by the end of 30 days. Understanding exponential functions is essential for grasping many phenomena in science, biology, and economics, where the growth isn't linear but rather exponential. This mathematical tool allows us to project future outcomes based on current data and growth patterns.