Chapter 4Notes for 20 August

Notes also available as PDF.

4.1 Review: Inductive and deductive reasoning

Inductive
making an “educated” guess from prior observations.
Deductive
if premises are satisfied, conclusion follows.

Mathematics is a combination of both forms of reasoning in no particular order.

• Problems to find: inductive
• Problems to prove: deductive
• Finding a proof… both!

Recall examples:

• Example of inductive reasoning: Extending a sequence from examples.
• Example of deductive reasoning: Deriving a rule for computing a sequence.

Always take care with your premises. Be sure you understand the framework before exploring with guesses.

4.2 Inductive reasoning on sequences

Purpose: Define some terminology. See how different sequences grow.

Sequence
list of numbers
Term
one of the numbers in a list

Examples:

• 3, 5, 7, 9, 11, …
• 4, 12, 36, 108, …

(Elipsis is three dots and is not followed by a comma. Text’s use is incorrect on page 10.)

Two common types of sequences:

Arithmetic
Defined by an initial number and a constant increment.
Geometric
Defined by an initial number and a constant multiple.

In our examples:

• 3, 5, 7, 9, 11, … : Arithmetic, starts with 3, incremented by 2.
• 4, 12, 36, 108, … : Geometric, starts with 4, multiplied by 3.

On growth:

• Note how the arithmetic sequence’s growth is “smooth”, linear.
• The geometric sequence grows much more quickly, exponential.

4.3 A tool for sequences: successive differences

Technique is useful for finding an arithmetic sequence buried in a more complicated appearing sequence of numbers.

This is an example of reducing to a known, simpler problem. We will explore this and other general problem solving methods shortly.

Simple example with an arithmetic sequence:

 3 5 2 7 2 9 2 11 2

Note that the last column provides the increment.

Another example, not directly arithmetic:

 2 6 4 22 16 12 56 34 18 6 114 58 24 6

The third column is an arithmetic sequence.

To obtain the next term, fill in the table from the right:

 2 6 4 22 16 12 56 34 18 6 114 58 24 6 202 88 30 6

4.4 Successive differences are not useful for everything.

What if we apply this to the geometric sequence above?

 4 Completing the 12 8 table is 36 24 16 not necessary. 108 72 48 Look at the 324 216 144 growth 972 648 432
• Note that each successive column grows just as quickly as the first.
• Divide the first column by 4, second by 8, etc., and what happens? The columns are the same.
• Successive differences of a geometric sequence still are geometric sequences.

4.5 An application where successive differences work, amazingly.

• Will return to the “number patterns” examples in the future.
• Skipping to the “figurate numbers” as another example of successive differences.
• Also to define common terminology.

For the terminology, consider the following table header from the context of sequences:

 n {T}_{n} {S}_{n}
• In general, n in mathematics is an integer that counts something.
• Here, the term (individual number) within a sequence (list of numbers).
• n = 1 is the first term, n = 2 the second, etc.
• A sequence often is named with a letter. Here T and S for triangular and square. Will explain the names in a moment.
• A particular term n in sequence T is {T}_{n}.

To explain the names, start with two points. Draw triangles off of one, squares off the other. Fill in the following table:

 n {T}_{n} {S}_{n} 1 1 1 2 3 4 3 6 9 4 10 16 5 15 25

The text provides formula. Plug in n, get a number. Or apply successive differences:

 n {S}_{n} {Δ}_{n}^{(1)} = {S}_{n} - {S}_{n-1} {Δ}^{(2)} = {Δ}_{n}^{(1)} - {Δ}_{n-1}^{(1)} 1 1 2 4 3 3 9 5 2 4 16 7 2 5 25 9 2

Terminology notes: A superscript with parenthesis often indicates a step in a process. Here it’s the depth of the difference. And Δ (Greek D, “delta”) is a traditional letter for differences.

4.7 Homework

Practice is absolutely critical in this class.

Groups are fine, turn in your own work. Homework is due in or before class on Mondays.

• Exercises for Section 1.1:
• Even problems 2-12. One short sentence of your own declaring why you decide the reasoning is inductive or deductive. Feel free to scoff where appropriate.
• Explain why the Section 1.1’s example of “2, 9, 16, 23, 30” is a trick question.
• Exercises for Section 1.2:
• Problems 2, 9, and 10.
• Problems 14 and 16.
• Problems 29 (appropriate formula is above problem 21), and 30.
• Problems 32, 39, and 51.

Note that you may email homework. However, I don’t use MicrosoftTM products (e.g. Word), and software packages are notoriously finicky about translating mathematics.

If you’re typing it (which I advise just for practice in whatever tools you use), you likely want to turn in a printout. If you do want to email your submission, please produce a PDF or PostScript document.