Sequences and series

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

2. 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. If we let the ball bounce twice, it would rebound a
total distance of 4 + .
(a) What is the total distance that the golf ball rebounds if you drop it from 6
feet, and watch it rebound successively 10 times? (3 marks)
(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? (2
marks)

3. (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.

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

5. 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

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. 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

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

9. 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?