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# Pre-Cal 40S June 8, 2009

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Sequences workshop.

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### Pre-Cal 40S June 8, 2009

1. 1. So much math in one picture. Can you see it? Get into 7 groups. You guys know what to do. I'll be back as soon as I get the Applied Math Prov Exam Started. Gaby fountain drosted by ﬂickr user ocelotan
2. 2. The sum of the terms in an inﬁnite geometric series is 4 and the common ratio is . Find the ﬁrst term.
3. 3. A super ball is dropped from a height of 200 cm. It rebounds to ¾ of the distance it fell each time it hits the ground. What is the total vertical distance traveled by the ball when it hits the ground for the fourth time? SooperBalls! by ﬂickr user Ben McLeod
4. 4. A super ball is dropped from a height of 200 cm. It rebounds to ¾ of the distance it fell each time it hits the ground. What is the total vertical distance traveled by the ball when it hits the ground for the fourth time? SooperBalls! by ﬂickr user Ben McLeod
5. 5. A super ball is dropped from a height of 200 cm. It rebounds to ¾ of the distance it fell each time it hits the ground. What is the total vertical distance traveled by the ball when it hits the ground for the fourth time? What is the total vertical distance the ball travels when it comes to a stop?
6. 6. The enrollment at DMCI was 400 in 1973. If the school’s population has increased 5% a year, how many students will be going to DMCI in 2010?
7. 7. If a sheet of paper 0.002 cm thick is torn in half 50 times , with all the pieces piled on top of each other prior to each tear, how thick is the stack of paper to the nearest km? Read about Britney Gallivan
8. 8. The Bouncing Ball EXTRA PRACTISE A ball is dropped from one metre, and the height is recorded after each bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90 percent of its drop height if it is dropped onto concrete from a height of less than two metres. 1. How high does the ball bounce on its eighth bounce? 2. How many times does the ball bounce before it rises to less than half of its original drop height? 3. How many times does the ball bounce before it stops bouncing? 4. How far has the ball travelled as it reaches the top of its 4th bounce. 5. Construct a graph that shows the bounce height versus bounce number.
9. 9. The Sierpinski Triangle Waclaw Sierpinski, a Polish mathematician, developed another fractal known as the Sierpinski Triangle. This fractal also starts with an equilateral triangle. To draw the fractal, you ﬁnd the midpoint of each side of the original triangle, and then draw three segments joining the midpoints. There are now four triangles inside the original triangle. The middle triangle is not shaded, and the process is continued with the other three shaded triangles, as shown in the diagram below.
10. 10. A Fractal: The Koch Snowﬂake All about the Koch Snowﬂake on wikipedia
11. 11. TED Talks Ron Eglash: African fractals, in buildings and braids http://www.ted.com/index.php/talks/view/id/198
12. 12. The fractal shown in the diagram below is created as follows: • A shaded triangle is formed by joining the midpoints of the vertical and horizontal sides. • A vertical line is drawn from the midpoint of the horizontal side, creating a new isosceles right triangle. • The process is continued. Find the total shaded area. Original 1st Iteration 2nd Iteration
13. 13. If the third term of a geometric sequence is 36 and the eighth term is 8748, ﬁnd the ﬁrst term.
14. 14. Given a geometric sequence with = 12 and = 4, ﬁnd .
15. 15. Suppose that a golf ball, when dropped on a ﬂoor, rebounds 2/3 of the distance from which it is dropped. For example, if the ball is dropped from 6 feet, the ball will bounce upwards 4 feet. (a) What is the total distance that the golf ball rebounds if you drop it from 6 feet, and watch it rebound successively 4 times? (b) What is the total distance that the golf ball rebounds if you drop it from 6 feet, and watch it rebound successively until it comes to a stop?
16. 16. (a) Explain why {8,4,2,0} cannot be the ﬁrst 4 terms of an arithmetic sequence. (b) Show how you can make {8,4,2,0} into the ﬁrst four terms of an arithmetic sequence by changing only one term. (c) Show how you can make {8,4,2,0} into the ﬁrst four terms of a geometric sequence by changing only one term.
17. 17. Two species of ants, the red ants and black ants, are preparing for battle. Each day the number of red ants increases by 2%, while the number of black ants increases by 2000 per day. Initially (day 1) each side has 1000 ants. (a) Find an explicit formula for the number of black ants on day n. (b) Find an explicit formula for the number of red ants on day n. (c) On day 365 which species has the larger population?