The Bouncing Ball

Working with Sequences on the TI-83+ or 84+

A good resource for learning your way around the calculator or to review what we've learned in class ...

http://www.mathbits.com/MathBits/TISection/PreCalculus/sequences.htm

Geometic Sequence:

(ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by an exponential equation.

(i) Recursive Definition: An ordered list of numbers generated by continuously multiplying a value (the common ratio) with a given first term.

a = 3

r = 23, 6, 12, 24, 48, ...

3, 6, 12, 24, 48, ...t = 3(2)n(n - 1)

Common Ratio (r):

(ii) From the implicit definition, r is the base of the exponential function.

(i) The number that is repeatedly multiplied to successive terms in a geometic sequence.

a = 3

r = 2

t = 3(2)n(n - 1)

To Find The Common Ratio

t(n - 1) is the term immediately before tn in the sequence

tn is an arbitrary term in the sequence

r is the common ratio

3, 6, 12, 24, 48, ...

To Find the nth Term In a Geometic Sequence

r is the common ratio

n is the "rank" of the nth term in the sequence

a is the first term

tn is the nth term

3, 6, 12, 24, 48, ...

Once we know the pattern of a sequence, we can find any term of the sequence. Use your calculator to check your answers.

Find the 21st term of: 3, 6, 12, 24, ...

Find the 110th term of: 5, 8, 11, 14, ... HOMEWORK

U(110)= 332

U(21)= 3 145 728

Find the 27th term of: 1, 1, 2, 3, 5, 8, ... HOMEWORK

U(27)= 196 418

Which term in the sequence: 2, 5, 11, 23, 47, ... is 1535?HOMEWORK

My cat is sick. His name is Little John and he has a cold. The vet has given him some medicine. Each day he gets a pill with 35 mg of medicine. His body eliminates 25% of the medicine each day and then he gets another pill.

(b) Will the amount of medicine in his body stabilize? How many days will it take and how much medicine will be in his body?

HOMEWORK

My cat is sick. His name is Little John and he has a cold. The vet has given him some medicine. Each day he gets a pill with 35 mg of medicine. His body eliminates 25% of the medicine each day and then he gets another pill.

(b) Will the amount of medicine in his body stabilize? How many days will it take and how much medicine will be in his body?

(a) How much medicine will be in his body in 5 days?

HOMEWORK

A small forest of 4000 trees is under a new forestry plan. Each year 20% of the trees will be harvested and 1000 new trees are planted.

(b) Will the forest size ever stabilize? If so, how many years and with how many trees?

(a) Will the forest ever disappear?

A small forest of 4000 trees is under a new forestry plan. Each year 20% of the trees will be harvested and 1000 new trees are planted.

(b) Will the forest size ever stabilize? If so, how many years and with how many trees?

(a) Will the forest ever disappear?

Sequences and Series on YouTube

Introduction to today's class by Mr. Green on YouTube ... a summary of almost everything in this unit ...

http://youtube.com/watch?v=WjLSz-nNLBc

The Bouncing Ball 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?

5. Construct a graph that shows the bounce height versus bounce number.

4. How far has the ball travelled as it reaches the top of its 4th bounce.

3. How many times does the ball bounce before it stops bouncing?

2. How many times does the ball bounce before it rises to less than half of its original drop height?