Problem 105 - Essential Calculus

Question

Use the formula for the \(n\) th partial sum of a geometric series $$\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}$$ You go to work at a company that pays \(\$ 0.01\) for the first day, \(\$ 0.02\) for the second day, \(\$ 0.04\) for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working (a) 29 days, (b) 30 days, and (c) 31 days?

Step-by-Step Solution

Answer

The total income for working 29 days is approximately 5368.71 USD, for 30 days is approximately 10737.42 USD, and for 31 days is about 21474.83 USD.

Step 1: Identify the parameters

Identify the first term a and the common ratio r in the geometric series. Here, a = 0.01 USD (the wage on the first day) and r = 2 (each day's wage is double the previous day's).

Step 2: Apply the formula for a 29-day work period

Plug the parameters into the geometric series partial sum formula and calculate the sum for n = 29. This gives the total income for working 29 days: \[S = \frac{a(1 - r^n)}{1 - r} = \frac{0.01(1 - 2^{29})}{1 - 2}.\]

Step 3: Apply the formula for a 30-day work period

Repeat this process for n = 30 to find the total income for working 30 days: \[S = \frac{a(1 - r^n)}{1 - r} = \frac{0.01(1 - 2^{30})}{1 - 2}.\]

Step 4: Apply the formula for a 31-day work period

Finally, repeat the process once more for n = 31 to find the total income for working 31 days: \[S = \frac{a(1 - r^n)}{1 - r} = \frac{0.01(1 - 2^{31})}{1 - 2}.\]

Key Concepts

Partial Sum

The partial sum is a term often used in relation to geometric series. When dealing with a geometric series, the partial sum refers to the sum of a specified number of initial terms. In this case, if you're working for a company that increases your pay daily, knowing the partial sum helps you calculate your total earnings over a period. For instance, if you want to know your cumulative salary over 29 days, you would calculate the partial sum of the first 29 terms in the series.

• It helps in finding the total accumulated value over a sequence of periods.
• Partial sums are essential for calculating values in real-world applications, such as salaries or loans that follow a geometric progression.

This process involves summing up the earnings of each day up to the desired point, using the formula for the sum of a geometric series.

Common Ratio

The common ratio in a geometric series is the factor by which we multiply each term to get the next term. In the given scenario, the common ratio is 2, meaning each day's wage is double that of the previous day. Understanding the common ratio is crucial in predicting and calculating future terms of the series. Here’s why the common ratio matters:

• It helps determine the pattern in which your values grow. Here, your salary grows exponentially due to a common ratio of 2.
• The common ratio is uniform, which makes calculations straightforward once it's identified.

In geometric sequences, the operations are largely based around this factor, making it a fundamental concept to grasp.

First Term

The first term of a geometric series, denoted as \(a\), is the starting point of the sequence. It's your initial value, which in this context is the first day's wage, \(0.01\) USD. Knowing the first term is critical because it sets the base for your calculations.Let's break it down:

• The first term \(a\) serves as the foundation on which the entire series is built.
• It's the value that compound growth through the common ratio starts from.

Without a defined first term, it becomes impossible to compute further terms in the series, making it an essential component in geometric progressions.

Mathematical Formula

A mathematical formula in the context of geometric series provides a systematic method for calculating sums or terms. The formula for the partial sum \(S_n\) of a geometric series is:\[S_n = \frac{a\left(1-r^n\right)}{1-r}\]Using this formula, you can easily calculate your total income over any given number of days:

• \(a\) is the first term of the series.
• \(r\) is the common ratio.
• \(n\) is the number of terms you want to sum up (in our example, the number of days worked).

By plugging in the appropriate values, you can compute how much you will earn over a specific period, thanks to the formula's structured approach. This highlights the utility of mathematical formulas in providing efficient solutions to complex problems.