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.
Which option would you choose? Defend your decision mathematically.
How much money would you have at the end of the 7 days if you choose each option?
Describe each option as linear, quadratic or exponential. Prove your description using finite differences.
State two reasons why "money isn't everything" when it comes to a job.
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To determine which pay option to choose, we can calculate the total amount of money earned for each option and compare them. Let's go through each option and calculate the amount of money earned:
Option 1:
In this option, you are paid $1 on the first day and $2 more each day that follows. So, the total amount of money earned for each day can be represented by the arithmetic sequence: 1, 3, 5, 7, 9, 11, 13.
To find the total amount of money earned, we need to sum up this sequence. The formula to find the sum of an arithmetic sequence is:
Sum = (n/2) * (first term + last term), where n is the number of terms.
In this case, n = 7 (for 7 days), the first term is 1, and the last term is 13. Plugging in these values, we get:
Sum = (7/2) * (1 + 13) = 7 * 14 = 98.
So, if you choose Option 1, you would have $98 at the end of the 7 days.
Option 2:
In this option, you are paid $1 the first day, then $4 the next day, $9 the day after that, and so on. The amounts of money earned form the sequence: 1, 4, 9, 16, 25, 36, 49.
To find the total amount of money earned, we can again use the formula for the sum of a sequence. Plugging in the values, we get:
Sum = (7/2) * (1 + 49) = 7 * 50 = 350.
So, if you choose Option 2, you would have $350 at the end of the 7 days.
Option 3:
In this option, you are paid $1 the first day, then double that amount every day. The amounts of money earned form the sequence: 1, 2, 4, 8, 16, 32, 64.
Again, we can use the sum formula to find the total amount of money earned:
Sum = (7/2) * (1 + 64) = 7 * 65 = 455.
So, if you choose Option 3, you would have $455 at the end of the 7 days.
Now let's describe each option mathematically and prove it using finite differences:
Option 1 is a linear sequence because the difference between consecutive terms is constant (2). The first finite difference is constant at 2.
Option 2 is a quadratic sequence because the difference between consecutive terms is increasing linearly (3, 5, 7, 9, ...). The second finite difference is constant at 2.
Option 3 is an exponential sequence because the ratio between consecutive terms is constant (2). The ratio is evident in the pattern of multiplying the previous term by 2.
Now, why is "money isn't everything" when it comes to a job? Here are two reasons:
1. Job Satisfaction: Money alone does not necessarily guarantee job satisfaction. Factors like work-life balance, the fulfillment of your tasks, room for growth, and a positive work environment can significantly impact your overall job satisfaction.
2. Work-Life Balance: Sometimes, jobs that pay more require longer working hours or involve high-stress levels. Prioritizing a healthy work-life balance can be more important to some individuals, even if it means taking a job with slightly lower pay.
It's important to consider not only the monetary aspect but also personal preferences, long-term goals, and job satisfaction when making decisions about employment.