Sequences workshop.

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. The sum of the terms in an infinite geometric series is 4 and the common
ratio is . Find the first term.

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. 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. 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. 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. 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. 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. 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. A Fractal: The Koch Snowflake
All about the Koch Snowflake on wikipedia

11. TED Talks Ron Eglash: African fractals, in buildings and braids
http://www.ted.com/index.php/talks/view/id/198

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. If the third term of a geometric sequence is 36 and the eighth term is
8748, find the first term.

14. Given a geometric sequence with = 12 and = 4, find .

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. (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. 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?