1 / 33

Complete the Patterns:

Complete the Patterns:. Patterns and Inductive Reasoning. (notes). Find and describe patterns. Inductive Reasoning. Making a conclusion based on a pattern of examples or past events. Example 1 : Find the next 3 terms of the sequence. 33, 39, 45, ….

jorryn
Download Presentation

Complete the Patterns:

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Complete the Patterns:

  2. Patterns and Inductive Reasoning (notes) Find and describe patterns

  3. Inductive Reasoning Making a conclusion based on a pattern of examples or past events.

  4. Example 1: Find the next 3 terms of the sequence. 33, 39, 45, … I’ll look at adding or subtracting the numbers 1st. Answer: 51, 57, 63 (add 6)

  5. Example 2: Find the next figure in the pattern. Answer: Look at the colors and that dot.

  6. Use Inductive Reasoning Inductive Reasoning * Look for a Pattern * Make a Conjecture based on your observations * Verify the Conjecture using logical reasoning

  7. Conjecture A conclusion that you reach based on observations (a pattern). Conjecture is like an educated guess. For example, if a number of dark clouds cover the sky and the wind picks up, one might conjecture that … It might rain

  8. An important part of a conjecture is that they areNOT always correct.    For example, after losing a lot of money in the slot machines, a person is likely to say, "I will win the next time" .... unfortunately this conjecture is usually wrong.

  9. It only takes 1 false example to show that a conjecture is not true. Counterexample Example 4: Find a counterexample for these statements… All dogs have spots. All prime numbers are odd.

  10. Learning Goal 3Sequences

  11. Sequence A sequence is an ordered set of numbers. 3, 5, 7, 9,… Each # is called a term of the sequence.

  12. an n 1 3 2 5 3 7 4 9 5 11 Sequence A sequence is really a function whose domain (n-values) are the positive integers. Index Number: tells you where you are in a sequence Term: the numbers or variables of the sequence

  13. Infinite sequence goes on forever. More #s in the sequence Finite sequence stops.

  14. Infinite or Finite If the sequence goes on forever it is called an infinite sequence, otherwise it is a finite sequence Identify if the following are infinte or finite. Then state the rate of change. infinite sequence Examples 1, 2, 3, 4 ,... +1 infinite sequence +5 20, 25, 30, 35, ... finite sequence +2 3, 5, 7, 9 infinite sequence x2 1, 2, 4, 8, 16, 32, ...

  15. Ex. 1 Find the first four terms of the sequence This is just like… f(n) = 3n – 2

  16. Ex. 1 Find the first four terms of the sequence a1 = 3(1) - 2 = 1 First term a2 = 3(2) - 2 = 4 Second term a3 = 3(3) - 2 = 7 Third term a4 = 3(4) - 2 = 10 Fourth term Find the 10th term and the 15th term of the sequence

  17. Recursive sequences • Each term in a recursive sequence is defined by the term before it.

  18. RECURSIVE FORMULA • Look for a pattern in how each term relates to the previous term

  19. How to write a Recursive Formula: • Step One: declare the first term • Step Two: what ever operation you do to get the next term…write it as an equation a1 =_____ an = an-1 + ____

  20. YOU TRY: Write a Recursive Formula: Step One: declare the first term Step Two: what ever operation you do to get the next term…write it as an equation a1 =_____ an = an-1 + ____ 3 2

  21. Find the first five terms of this recursively defined sequence: Ex. 3 a1 = 1 and an = 3(an-1) - 1

  22. Find the first five terms of this recursively defined sequence: UTRY a1 = 4, an = a n-1 - 2

  23. Writing the formula…or CLOSED FUNCTION • We may know the first few terms of a sequence, but not the formula. • In such a case, we look for a pattern. • Relate the term to the position that it is in.

  24. The Rule A Sequence will have a Rule that gives you a way to find the value of each term. Example: the sequence 3, 5, 7, 9, ... starts at 3 and jumps 2 every time: 

  25. But The Rule Should be a Formula! But, saying "starts at 3 and jumps 2 every time" doesn't tell us how to calculate the: 10th term, 100th term, or nth term (where n could be any term number we want). That would be A LOT of counting! So, we want a formula with "n" in it (where n can be any term number that we use to SOLVE!)

  26. So, What Is The Rule For 3, 5, 7, 9, ...? First, we can see the sequence goes up 2 every time, so we can guess that the rule will be something like “n + 2” (where "n" is the term number). Let's test it out! Test Rule: n+2 NOPE!

  27. So, What Is The Rule For 3, 5, 7, 9, ...? Since that didn’t work, let’s try multiplying 2 by each term number, n like: "2 × n" Let's test it out! Test Rule: 2n NOPE!

  28. That nearly worked ... but that Rule is too low by 1 every time, so let us try changing it to: Test Rule: 2n+1 That Works!

  29. So instead of saying "starts at 3 and jumps 2 every time" we write the rule as: The Rule for 3, 5, 7, 9, ... is: an = 2n+1 The Rule is in Closed Form. How did we do that? There’s a simple formula you can use EVERY time to help you figure it out! an = mn + b

  30. an n 1 1 2 3 3 5 4 7 Write the formula. Ex. 5 1, 3, 5, 7, 10… 1st, 2nd, 3rd, 4th, an = 2n – 1

  31. Now, if we want to calculate the 10th term we can write: an = 2n+1 a10= 2(10)+1 a10= 21 We can also calculate the 100th term: n = 100 2(100) + 1 The 100th term is 201 Can you calculate the 50th term? The 500th term?

  32. QUESTIONS? What are your Questions?

  33. Class Work/Homework • Sequences worksheet

More Related