Suppose you go to a company that pays $0.04 for the first day, $0.08 for the second day, $0.16 for the third day, and so on. If the daily wage keeps doubling, what will your total income be for working 31 days? Total Income = $
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Suppose you go to a company that pays $0.04 for the first day, $0.08 for the second day, $0.16 for the third day, and so on.
![### Doubling Daily Wage Problem
**Problem Statement:**
Suppose you go to a company that pays $0.04 for the first day, $0.08 for the second day, $0.16 for the third day, and so on.
**Question:**
If the daily wage keeps doubling, what will your total income be for working 31 days?
**Solution:**
To find out the total income for working 31 days, we need to sum up the amount paid each day for all 31 days. The wage pattern follows a geometric sequence:
- Day 1: $0.04
- Day 2: $0.08
- Day 3: $0.16
- ...
The daily wage can be represented as \( 0.04 \times 2^{n-1} \), where \( n \) is the day number.
To find the total income over 31 days, we sum the wages for each day:
\[ \text{Total Income} = 0.04 \times (2^0 + 2^1 + 2^2 + \cdots + 2^{30}) \]
This is a geometric series with the first term \( a = 0.04 \) and common ratio \( r = 2 \). The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
For our problem, \( n = 31 \), \( a = 0.04 \), and \( r = 2 \):
\[ S_{31} = 0.04 \times \frac{2^{31} - 1}{2 - 1} \]
\[ S_{31} = 0.04 \times (2^{31} - 1) \]
**Calculated using:**
\[ S_{31} = 0.04 \times (2147483648 - 1) \]
\[ S_{31} = 0.04 \times 2147483647 \]
\[ S_{31} = 85899345.88 \]
So, the total income for working 31 days is:
\[ \text{Total Income} = \$85,899,345.88 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07450324-4898-4505-8a10-20a625d5cb7d%2Fba184dd9-dd41-4a1a-93b7-2aee62d4370e%2Fegicbmi.png&w=3840&q=75)
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