2. The Penny Problem Your parents have decided that since you are becoming more independent, you should have your own money. They are going to give you some money and offer you two options $$$ OPTION 1 $$$ They will give you: $10,000 today, but nothing else for the rest of the year $$$ OPTION 2 $$$ They will give you: Day 1: $0.01 (one penny) Day 2: $0.02 Day 3: $0.04 Day 4: $0.08 They will continue this pattern for one month (30 days). After that you get nothing the rest of the year.

3. The Penny Problem Copy the table to the right. Fill out the blanks by following the pattern Day #Paid on DayPaid Total 1$0.01 2$0.02$0.03 3$0.04$0.07 4$0.08$0.15 5 6 7 8 9

4. The Penny Problem Which group won? Did the amount of money surprise you? How do you think it got so big? How does this pattern compare to the patterns we looked at before?

5. Sequences Let ’ s back up What is an Arithmetic Sequence? How does the pattern “ work ” ? How do the numbers “ grow ” ? ( hint: recursive rule ) What happened in the table? What did the graph look like? Before today we were talking about “ Arithmetic Sequences. ” Let ’ s review what these are.

6. Sequences Let ’ s compare… “ Yes, the Penny Problem was an Arithmetic Sequence ” If you think this is an arithmetic sequence, why do you think so? “ No, the Penny Problem was NOT an Arithmetic Sequence ” If you think this is some other kind of pattern, why do you think so? Was the Penny Problem pattern a Arithmetic Sequence?

7. Sequences The Penny Problem is not an Arithmetic Sequence Have we seen a pattern like this before? The pattern in the Penny Problem is called a GEOMETRIC SEQUENCE Can you come up with the recursive rule for this pattern? This might help… D=10.01D=50.16 D=20.02D=60.32 D=30.04D=70.64 D=40.08D=81.28

8. Sequences The Fractals were also a Geometric Sequence Sierpinski ’ s Triangle Fractal Quilt Stage012345 Triangles1392781243 Stage012345 Triangles15251256253125 Penny Problem Stage123456 Amount$0.01$0.02$0.04$0.08$0.16$0.32 Each of these Geometric Sequences has something in common. Can you figure it out? ( hint: recursive rule )

9. Sequences Remember… The bridges problem showed an Arithmetic Sequence Remember our Arithmetic Sequences? Sections1234 Beams371115 We looked at a couple others earlier… Phase1234 Hearts25811 Phase1234 Cookies3456 Each of these Arithmetic Sequences has something in common. Can you figure it out? ( hint: recursive rule )

10. Sequences How are Arithmetic Sequences and Geometric Sequences different? FIRST Write down this question and answer it on your own. THEN After you have your answer Compare your answer with another student in the room No need to get up if you have people sitting right next to you!

11. Sequences How are Arithmetic Sequences and Geometric Sequences different? Arithmetic Sequences grow through repeated addition. A recursive rule in an Arithmetic Sequence will often look like: “ Add some value to the previous term. ” Geometric Sequences grow through repeated multiplication. A recursive rule in a Geometric Sequence will often look like: “ Multiply the previous term by some value. ”

12. 1) 2, 6, 10, 14, ____ 2) 2, 6, 18, 54, ____ 3) 7, -14, 28, -56, ____ 4) 26, 21, 16, 11, ____ 5) 36, 18, 9, 4.5, ____ 6) -8, -1, 6, 13, ____ Geometric OR Arithmetic? What is the rule? Finish the pattern.