Transcript Sequence - Edublogs

Chapter
1
An Introduction to
Problem Solving
1-2 Explorations with Patterns
 Inductive Reasoning
 Arithmetic Sequences
 Fibonacci Sequence
 Geometric Sequences
 Other Sequences
EXAMPLE 1-1
Describe any patterns:
There are several possible patterns. For
example, the numbers on the far left are natural
numbers. The pattern starts with 1 and continues
to the next greater natural number in each
successive line.
EXAMPLE 1-1
(continued)
The numbers “in the middle” are products of two
numbers, the second of which is 9. The first
number in the first product is 0; after that the first
number is formed using natural numbers and
including one more in each successive line.
EXAMPLE 1-1
(continued)
The resulting numbers on the right are formed
using 1s and include an additional 1 in each
successive line.
EXAMPLE 1-1
(continued)
Do the patterns continue? Why or why not?
The pattern in the complete equation appears to
continue for a number of cases, but it does not
continue in general; for example,
This pattern breaks down when the pattern of
digits in the number being multiplied by 9
contains previously used digits.
Inductive Reasoning
 Make generalizations based on
observations and patterns.
 May lead to a conjecture – a statement
that is thought to be true but hasn’t been
proved.
 Can be proven false by a counterexample
– a statement that contradicts the
conjecture.
The Danger of Making Conjectures
Choose points on a circle and connect them to
form distinct, nonoverlapping regions. In this
figure, 2 points determine 2 regions, 3 points
determine 4 regions, and 4 points determine 8
regions. What is the maximum number of regions
that would be determined by 10 points?
It appears that each time we increase the number
of points by 1, we double the number of regions. If
this were true, then for 5 points we would have 2
times the number of regions with 4 points, or
and so on.
If we base our conjecture on this pattern, we might
believe that for 10 points, we would have 29, or 512
regions.
An initial check for this conjecture is to see whether
we obtain 16 regions for 5 points.
For 6 points, the pattern predicts that the number of
regions will be 32.
There are actually 31 regions, so the conjecture is
not true.
Definition
Sequence: an ordered arrangement of numbers,
figures, or objects.
Sequences can be classified by their properties.
Arithmetic Sequences
A sequence in which each successive term is
obtained from the previous term by addition or
subtraction of a fixed number (the difference).
Example
The sequence 2, 5, 8, 11, 14, … is an arithmetic
sequence because the difference between each
term is 3.
Example 1-2
Find a pattern in the number of matchsticks
required to continue the pattern.
Note that the addition of a square to an
arrangement requires the addition of three
matchsticks each time. Thus, the numerical pattern
obtained is 4, 7, 10, 13, 16, 19, … , an arithmetic
sequence with a difference of 3.
Example 1-3a
Find the first four terms of a sequence given by
and determine whether the sequence
seems to be arithmetic.
Number of
Term
1
2
3
4
Term
4·1 + 3 = 7
4·2 + 3 = 11
4·3 + 3 = 15
4·4 + 3 = 19
The sequence
seems arithmetic
with difference 4.
Example 1-3b
Find the first four terms of a sequence given by
and determine whether the sequence
seems to be arithmetic.
Number of
Term
1
2
3
4
Term
12 − 1 = 0
22 − 1 = 3
32 − 1 = 8
42 − 1 = 15
The sequence is
not arithmetic
because it has no
common difference.
Example 1-4
The diagrams in the figure show the molecular
structure of alkanes, a class of hydrocarbons.
C represents a carbon atom and H a hydrogen
atom. A connecting segment shows a chemical
bond. (Remark: CH4 stands for C1H4.)
Example 1-4
(continued)
Hectane is an alkane with 100 carbon atoms. How
many hydrogen atoms does it have?
First study the drawing of the alkanes and
disregard the extreme left and right hydrogen
atoms in each. With this restriction, we see that for
every carbon atom, there are two hydrogen atoms.
Therefore, there are twice as many hydrogen
atoms as carbon atoms plus the two hydrogen
atoms at the extremes.
Example 1-4
(continued)
This can be summarized as follows.
Example 1-5
A theater is set up in such a way that there are
20 seats in the first row and 4 additional seats in
each consecutive row. The last row has 144
seats. How many rows are there in the theater?
There are 32 rows in the theater.
Fibonacci Sequence
The Fibonacci sequence is defined by
for n = 3, 4, 5, …
The sequence is:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Geometric Sequences
A sequence in which each successive term is
obtained from the previous term by multiplying by
a fixed nonzero number (the ratio).
Example
The sequence 2, 6, 18, 54, 162, … is a geometric
sequence because the ratio between each term is
3.
Definition
If n is a natural number, then
n factors
If n = 0 and a ≠ 0, then a0 = 1.
Other Sequences
Figurate numbers are
sequences of numbers
that can be
represented by dots
arranged in the shape
of geometric figures.
Triangular numbers
Square numbers
Example 1-6
Use differences to find a pattern. Then, assuming
that the pattern discovered continues, find the
seventh term in the sequence.
5
6
14 29 51 80
1
8
15 22 29
7
7
7
7
The first difference is an arithmetic sequence with
difference 7. The sixth term in the first difference
row is 29 + 7 = 36, and the seventh term in the
original sequence is 80 + 36 = 116.