Problem 8 - Applied Calculus

Question

An employer pays you 1 penny the first day you work and doubles your wages each day after that. Find your total earnings after working 7 days a week for (a) One week (b) Two weeks (c) Three weeks (d) Four weeks

Step-by-Step Solution

Answer

(a) \$1.27 (b) \$163.83 (c) \$20971.51 (d) \$2,696,478.71

Step 1: Calculate Daily Earnings

To find the daily earnings, we start with 1 penny on the first day. Each subsequent day, the earnings double. Represent the daily earnings for the first 7 days as a list: 1, 2, 4, 8, 16, 32, 64 (all in pennies).

Step 2: Convert to Dollars - One Week

Add the earnings for each of the 7 days: \(1 + 2 + 4 + 8 + 16 + 32 + 64 = 127\) pennies. Convert pennies to dollars by dividing by 100. Thus, for one week, the total is \$1.27.

Step 3: Extend Calculation to Two Weeks

For the second week, continue doubling: 128, 256, 512, 1024, 2048, 4096, 8192 (in pennies). Sum the second week: \(128 + 256 + 512 + 1024 + 2048 + 4096 + 8192 = 16256\) pennies. Add this to the first week: \(127 + 16256 = 16383\) pennies, or \$163.83.

Step 4: Extend Calculation to Three Weeks

For the third week, continue doubling: 16384, 32768, 65536, 131072, 262144, 524288, 1048576 (in pennies). Sum the third week: \(16384 + 32768 + 65536 + 131072 + 262144 + 524288 + 1048576 = 2080768\) pennies. Total for three weeks is \(16383 + 2080768 = 2097151\) pennies, or \$20971.51.

Step 5: Extend Calculation to Four Weeks

For the fourth week, continue doubling: 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728 (in pennies). Sum the fourth week: \(2097152 + 4194304 + 8388608 + 16777216 + 33554432 + 67108864 + 134217728 = 267550720\) pennies. Total for four weeks is \(2097151 + 267550720 = 269647871\) pennies, or \$2,696,478.71.

Key Concepts

Doubling Pattern

The concept of doubling is quite straightforward and fascinating. In a doubling pattern, the quantity starts with an initial value and then doubles at a regular interval. It's an exponential growth pattern, meaning it grows rapidly over time. In the given exercise, imagine you receive 1 penny on the first day of work. The next day, your pay doesn't increase by just a penny; it doubles to 2 pennies. On the third day, it doubles again to 4 pennies, and this continues each subsequent day.

• Starting with a small amount like 1 penny seems insignificant initially.
• As days pass and the doubling continues, the amount quickly grows large.
• This pattern showcases exponential growth, where the next value in the sequence is found by taking the previous value and multiplying it by 2.

Understanding this concept is vital. It shows how something seemingly small can become substantial in a short time through consistent doubling.

Compound Interest

While compound interest is intrinsically tied to financial growth, it shares a similar essence with the doubling pattern due to its exponential nature. In compound interest, the principal amount increases over time at a rate that can be compounded annually, semi-annually, quarterly, monthly, etc. Here are some key features of compound interest:

• The interest earned itself starts earning interest as time goes on.
• This generates a snowball effect where the amount grows larger and larger exponentially.
• Compound interest is represented by the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where:
  • \(A\) is the amount of money accumulated after n years, including interest.
  • \(P\) is the principal amount (initial amount).
  • \(r\) is the annual interest rate (decimal).
  • \(n\) is the number of times that interest is compounded per year.
  • \(t\) is the time the money is invested for in years.

Just like with the doubling pattern, the small starts can lead to surprisingly large results. Compounded over time, even modest investments can grow significantly.

Sequence and Series

Sequences and series form the backbone for understanding patterns such as doubling or compound interest. In mathematics, a sequence is an ordered list of numbers and a series is the sum of a sequence of terms. For the exercise in question, the sequence begins with 1 (one penny) and continues to grow by following a predictable pattern, doubling each term.

• The sequence in the first week is: 1, 2, 4, 8, 16, 32, 64.
• As you sum these, you form a series: \[ S = 1 + 2 + 4 + 8 + 16 + 32 + 64 \]
• This pattern showcases a geometric series where each term after the first is the previous term multiplied by a common ratio, here it is 2.

Such sequences and series are commonly seen in finance and science for modeling and predicting behavior. Understanding the sum of sequences is essential for figuring out final totals, like your earnings over time. It also illustrates how each part of the sequence contributes to the cumulative total in a rapid way.