Lesson

- To visualize arithmetic and geometric sequences in real life.
- To apply knowledge about arithmetic and geometric sequences to choose the most effective way to save money.

- Paper
- Pens of different colors
- Calculator
- Internet

Sequences and series can be used to describe many real-world situations. One major application of sequences and series is in finances. In this investigation, we will investigate two different methods of saving money from a paycheck.

- Decide on an item you would like to save up your money for. This item should be worth at least $200$200 dollars.
- Suppose you have a job that pays you $100$100 dollars a week. When you get paid at the end of each week you add a certain amount of money to your savings account.
- For the first money-saving method, decide on an amount of money that you would like to save from your paycheck each week. You can save up to half of your paycheck.
- Create a table to show how much money you have at the end of each week using this saving method. Stop this table when you have reached or exceeded the amount of money you need to purchase your chosen item.
- For the second saving method, decide on an amount of money that you would like to save from your paycheck in the first week. You can choose an amount of up to $10$10 dollars.
- Each week, you double the amount of money you save. So if you chose to put away $6$6 dollars in week one, then at the end of week two you will add $12$12 more dollars to the savings account and so on.
- Create a table to show how much money you have at the end of each week using this saving method. Stop this table when you have reached or exceeded the amount of money you need to purchase your chosen item.

- Which saving method saved money faster? Why?
- What type of sequence does the first saving method represent? How do you know?
- What type of sequence does the second saving method represent? How do you know?
- Create a graph of the sequence generated by the first saving method.
- On the graph that you just created, graph the sequence generated by the second saving method in a different colored pen.
- Find the sum you are adding into the savings amount each week in the second saving method. Can you afford to add that money every week with your earnings of $100$100 dollars a week?
- How much money would you need to put into your savings on week 6 using the first savings method? Is this plausible given you make $100$100 dollars a week?
- How much money would you need to put into your savings on week 6 using the second savings method? Is this plausible given you make $100$100 dollars a week?
- Which saving method would you choose? Explain.
- Compare with a friend. What method did they choose? Why did they choose that plan? How many weeks of that savings plan did it take them to afford their chosen item?

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.