Download Unit 1.2

UNIT 1 - ARITHMETIC & GEOMETRIC SEQUENCES
Task #2 – Bacterial Growth (Geometric Sequences)
Common Core: HS.F-IF.3, HS.F-BF.1a, 2, HS.F-LE.2
MA40: ALGEBRA 2
Name:
Period:
INVESTIGATION
Bacterial Growth Problem: Consider the growth sequence of bacteria
cells if a cut by a rusty nail puts 25 bacteria cells into a wound and then
the number of bacteria doubles every quarter hour.
1.
Sketch a series of pictures that illustrates what is happening in this
problem. Then, using words, describe the situation.
Pictures:
Words:
2.
Complete the table.
# of
Quarter
Hours
0
1
2
3
4
…
50
…
99
…
n
Bacteria
Count
Table Scratch Work:
3.
Is it possible to model the growth of bacteria, as described in this
problem, with an Arithmetic Rule? Explain.
4.
Consider the pattern described in the table. Write an equation that
describes the current number of bacteria based on the number of
bacteria present 15-minutes earlier. What type of rule is this?
Explain.
Unit 1.2 – Bacterial Growth (Geometric Sequences) – (continued)
5.
Consider the table again. Write an equation that describes the total
amount of bacteria present at any given time. What type of rule is this?
Explain.
DEVELOPING MATH CONCEPTS & TERMS
Geometric Sequence – A sequence where the ratio of any term to the
previous term is constant. The constant ratio is called the common ratio and
is denoted by r.
Decide whether the sequence is geometric, arithmetic, or neither. Identify
the common ratio/difference.
a)  3,  6,  12,  24,  48, ...
b) 7, 0,  7,  14,  21, ...
c) 2, 4, 16, 256, ...
d) 2,
e) 3,  9, 27,  81, 243, ...
f) 4, 15, 26, 37, 48, ...
2 2 2 2
, ,
, , ...
3 9 27 81
Find the first four terms of the sequence defined by the geometric rule.
a)
an  7  2
n 1
b)
a1  2,
an  3  an1
a1  _______________  ______
a1  _______________  ______
a2  _______________  ______
a2  _______________  ______
a3  _______________  ______
a3  _______________  ______
a4  _______________  ______
a4  _______________  ______
______,______,______,______,
sequence
______,______,______,_____,
sequence
Unit 1.2 – Bacterial Growth (Geometric Sequences) – (continued)
c) What makes these two sequences geometric? Both rules are geometric
rules. How can this be when they are both so different? Be specific.
TYING THINGS TOGETHER
Consider the Bacterial Growth problem discussed earlier.
6.
Since a geometric sequence of numbers is one in which each
number in the sequence is multiplied by a constant to get the next
number, explain why the sequence of bacteria counts is a geometric
sequence.
7.
Look at the equation you derived in problem #4.
a) Describe how changing the initial number of bacteria cells and
the growth rate affects the equation. Identify the rate of change.
b) Write an equation that describes the following. A starting
culture of 47 bacteria cells that doubles every half hour.
Unit 1.2 – Bacterial Growth (Geometric Sequences) – (continued)
8.
Look at the equation you derived in problem #5.
a) Describe how changing the initial number of bacteria cells and
the growth rate affects the equation. Identify the rate of change.
b) Write an equation that describes the following. A starting
culture of 47 bacteria cells that doubles every half hour.
9.
Summary:
Identify another real world occurrence that can be modeled using a
geometric sequence. Explain why you believe a geometric sequence
is the correct model.