You have just landed your first summer job and your boss has told you that you can choose from one of 3 pay structures for the first week (7 days) that you work there. And after the first week, your pay will be frozen at the final rate for the remainder of the summer.
• Option 1 - You are paid $1 on the first day, and then $2 more each day that follows (i.e., $3, $5, $7, ... for the rest of the 7 days.
• Option 2 - You are paid $1 the first day, then $4 the next day, then $9 the day after that, then $16, then $25, and so on for the rest of the 7 days.
• Option 3 - You are paid $1 the first day, then two times that amount on the second day ($2), then two times that amount the next day ($4), then $8, $16, and so on for the rest of the week.
1. Which option would you choose? Defend your decision mathematically.
2. How much money would you have at the end of the 7 days if you choose each option?
3. Describe each option as linear, quadratic or exponential. Prove your description using finite differences.
4. State two reasons why "money isn't everything" when it comes to a job.
It looks like this is your homework, not ours. Someone will be glad to check your answer.
Once you see the pattern, listing 7 numbers in each case is a trivial problem.
Option 1 - they tell you the pattern
Option 2 - square the numbers
Option 3 - double the previous number
1. To determine which option to choose, we can calculate the total amount of money earned at the end of 7 days for each option.
Option 1:
The amount earned on the first day is $1.
From the second day onwards, the amount earned is increasing by $2 each day.
So, the total amount earned after 7 days can be calculated using the formula for the sum of an arithmetic series:
Total amount = (Number of terms / 2) * (First term + Last term)
Number of terms = 7 (since it's a 7-day period)
First term = $1
Last term = $1 + (7 - 1) * $2 = $1 + 6 * $2 = $1 + $12 = $13
Total amount = (7 / 2) * ($1 + $13) = 3.5 * $14 = $49
Option 2:
The amount earned on the first day is $1.
From the second day onwards, the amount earned is increasing by $3 each day.
So, the total amount earned after 7 days can be calculated using the sum of the squares formula:
Total amount = (Number of terms / 6) * [First term + Last term + (Number of terms - 1) * Common difference]
Number of terms = 7 (since it's a 7-day period)
First term = $1
Last term = $1 + (7 - 1)^2 = $1 + 6^2 = $1 + 36 = $37
Common difference = $3
Total amount = (7 / 6) * [$1 + $37 + (7 - 1) * $3] = (7 / 6) * [$1 + $37 + 6 * $3] = (7 / 6) * [$1 + $37 + $18] = (7 / 6) * $56 = $98
Option 3:
The amount earned on the first day is $1.
From the second day onwards, the amount earned is doubling each day.
So, the total amount earned after 7 days can be calculated using the formula for the sum of a geometric series:
Total amount = First term * [(Common ratio ^ Number of terms) - 1] / (Common ratio - 1)
First term = $1
Common ratio = 2
Number of terms = 7 (since it's a 7-day period)
Total amount = $1 * [(2^7) - 1] / (2 - 1) = $1 * [128 - 1] = $1 * 127 = $127
2. If you choose Option 1, you will have $49 at the end of 7 days.
If you choose Option 2, you will have $98 at the end of 7 days.
If you choose Option 3, you will have $127 at the end of 7 days.
3. Option 1 is a linear pay structure as the amount earned increases by a constant $2 each day.
Option 2 is a quadratic pay structure as the amount earned increases by a square number each day.
Option 3 is an exponential pay structure as the amount earned doubles each day.
To prove these descriptions, we can examine the differences between consecutive terms for each option.
For Option 1, the differences between consecutive terms are constant ($2).
For Option 2, the differences between consecutive terms are increasing by 2 each time (1^2, 2^2, 3^2, ...).
For Option 3, the differences between consecutive terms are doubling each time (2^1, 2^2, 2^3, ...).
4. Two reasons why "money isn't everything" when it comes to a job:
a) Job satisfaction and fulfillment: Money alone may not bring happiness or job satisfaction. Factors such as the nature of the work, personal growth opportunities, work-life balance, and a supportive work environment can contribute significantly to one's overall job satisfaction.
b) Personal values and priorities: Different individuals have different values and priorities in life. For some, a higher salary may be less important compared to other aspects like flexibility, meaningful work, or the opportunity to make a positive impact on the world. Ultimately, it's important to consider one's personal goals and align them with the job's overall fit, not just monetary compensation.