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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, ….

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Complete the Patterns:

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  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

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