Notes for 20 August

- 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.

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.

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 |

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.

- 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.

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 Microsoft^{TM} 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.