Chapter 4
Notes for 20 August

Notes also available as PDF.

4.1 Review: Inductive and deductive reasoning

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

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

Recall examples:

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.

list of numbers
one of the numbers in a list


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

Two common types of sequences:

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

In our examples:

On growth:

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:


Note that the last column provides the increment.

Another example, not directly arithmetic:

6 4

The third column is an arithmetic sequence.

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

6 4

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
324216144    growth

4.5 An application where successive differences work, amazingly.

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


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

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.

4.6 Next time: Problem solving techniques.

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.

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.