Survey of Mathematical Ideas Math 100 Chapter 1

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Survey of Mathematical IdeasMath 100Chapter 1

• John Rosson
• Tuesday January 23, 2007

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Chapter 1The Art of Problem Solving

• Solving Problems by Inductive Reasoning
• Number Patterns
• Strategies for Problem Solving
• Calculating, Estimating and Reading Graphs

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Number Patterns Successive Differences
The method of successive difference tries to find
a pattern in a sequence by taking successive
differences until a pattern is found and then
working backwards.
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Number Patterns Successive Differences
Fill in the obvious pattern and work backwards by
adding.
The method of successive differences predicts
3992 to be the next number in the sequence.
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Number Patterns Successive Differences
The method of successive differences is not
always helpful. Consider
1 2 3 5 8 13 21 34 55 89 144 233 377
610 987 1597 .
1 1 2 3 5 8 13 21 34 55 89 144
233 377 610 .
0 1 1 2 3 5 8 13 21 34 55 89
144 233 .
1 0 1 1 2 3 5 8 13 21 34
55 89 .
Since the sequence reproduces itself after
applying successive differences, the method can
give us no simplification.
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Number Patterns Sums
We can use patterns to conjecture formula for
extended sums.
Conjecture The sum of the first n numbers
True
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Number Patterns Sums
A little more interesting is the sum of squares
Conjecture The sum of the squares of the first n
numbers
True
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Number Patterns Sums
Consider the patterns that we have seen.
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Number Patterns Sums
Check some cases.
So 2 is a counterexample (as is 3) and the
conjecture is false. In this case close is not
enough
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Number Patterns Sums
Looking at the sums of cubes, the true conjecture
is .
True
The sums of higher powers have no simple formula.
They are formulated in terms of a new idea the
Bernoulli polynomials.
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Number Patterns Figurate Numbers
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Number Patterns Figurate Numbers
The figurate numbers are a classical source of
number sequences.
Triangular numbers 1, 3, 6, 10, 15, 21, 28, 36,
45, 55
Square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81,
100
Pentagonal numbers 1, 5, 12, 22, 35, 51, 70, 92,
117, 145
Hexagonal numbers 1, 6, 15, 28, 45, 66, 91, 120,
153, 190
Heptagonal numbers 1, 7, 18, 34, 55, 81, 112,
148, 189, 235
Octagonal numbers 1, 8, 21, 40, 65, 96, 133,
176, 225, 280
Nonagonal numbers 1, 9, 24, 46, 75, 111, 154,
204, 261, 325
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Number Patterns Figurate Numbers
We can calculate the figurate numbers using
successive differences. Consider the nonagonal
numbers.
1 9 24 46 75 111 154 204 261 325
396
8 15 22 29 36 43 50 57 64
71
7 7 7 7 7 7 7 7
7
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Number Patterns Figurate Numbers
Formulas for the figurate numbers
Considering these formulas leads us to conjecture
a formula for a general N-agonal number
True
Note that this formula works for N3 and N4. It
even works for N2 (biagonal numbers).
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StrategiesPolyas Four Step Process

• Understand the problem
• Devise a plan
• Carry out the plan
• Look back and check

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Other Strategies

• Make a table or chart.
• Look for a pattern.
• Solve a similar simpler problem.
• Draw a sketch.
• Use inductive reasoning
• Solve an equation.

• Use a formula.
• Work backward.
• Guess and check
• Common sense (?)
• Look for a catch if the problem seems too easy
  or impossible.

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Example
How must one place the integers from 1 to 15 in
each of the spaces below in such a way that no
number is repeated and the sum of the numbers in
any two consecutive squares is a perfect square?
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Example
Maybe the problem is easy. Let us just try to
fill in the squares following the consecutives
sum to square rule. Start, say, with 1. This
playing with the problem is part of
understanding it.
Let us construct a table to see how things add
up.
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Example
The table tells us that we have to start with
either an 8 or a 9 since these two numbers can
only be paired with one other number.
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Example
The plan is now to start with either 8 or 9 see
if we can fill in the table. Most numbers have
only two choices for neighbors and one choice
will eliminate the next.
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Assignments 2.1, 2.2, 2.3, 2.4