PreCalculusChapter3MathematicalInduction.pptx

PRECAL MATHEMATICAL INDUCTION

1. Chapter 3 (Mathematical Induction)

2. A: Sequences and Series A sequence is a function whose domain is the set of natural numbers. Examples: 1.) 2.) 3.) 4.) 1, 1, 2, 3, 5, 8, 13, 21, … Fibonacci sequence 5.) 2, 3, 5, 7, 11, 13, 17, 19, 23, … Prime numbers  

3. A: Sequences and Series Series, on the other hand, is the sum of the terms in a sequence Examples: 1.) 2.)  

4. A : Sequences and Series Summation Notation -- Shorthand method of indicating the sum of the first terms or the n th partial term. If is a constant, then Proof:  

5. A : Sequences and Series Summation Notation -- Shorthand method of indicating the sum of the first terms or the n th partial term. b. If is a constant, then Proof:  

6. A : Sequences and Series Summation Notation -- Shorthand method of indicating the sum of the first terms or the n th partial term. c. Addition of summation notation Proof:  

7. A : Sequences and Series Evaluate        

8. A : Sequences and Series 2. Evaluate        

9. A : Sequences and Series 3. Evaluate          

10. A : Sequences and Series 4. An important skill in mathematics is to not be intimidated by a humongous problem and be able to work on smaller cases before doing the large or general case. Try evaluating this item from 2019 MMC Grade 10 Elimination Round. Below is the simplified sigma version, and the screenshot of the problem from the paper itself. Click once to reveal the answer.   Quite a large answer, right? Like, what is 2019 factorial? A typical scientific calculator can go only up to 69 factorial without declaring “Math error” (or some error). The solution is on the next slide, click again when ready to proceed.  

11. A : Sequences and Series   Let’s work on with the first two fractions first     Recall:     What now? Observe that:    

12. A : Sequences and Series   Current “accumulated sum”:        

13. A : Sequences and Series   Current “accumulated sum”:       Noticed something? For every succeeding fraction added to the “accumulated sum”, the sum increases to (the factorial of the denominator then minus one, all over the factorial of the denominator) You can continue by adding which will increase the sum to , adding the succeeding will give , and so on. By continuing in this pattern, the last fraction will be , then the sum (by pattern) will be  

14. A : Sequences and Series Arithmetic Progression -- is a sequence in the form of Where: is the first term is the n th term is the common difference Example: Arithmetic Means -- numbers inserted between the first term and last term of arithmetic progression Arithmetic Sum   Where: is the number of terms is the last term is the first term  

15. A : Sequences and Series Geometric Progression -- is a sequence in the form of Where: is the first term is the n th term is the common ratio Example: Geometric Means -- numbers inserted between the first term and last term of geometric progression Geometric Sum (Finite)   Where: is the number of terms is the last term is the first term   (Infinite)  

16. A : Sequences and Series 1 . Insert three arithmetic means between -3 and 12.              

17. A : Sequences and Series 2. Insert two geometric means between 4 and 256.              

18. A : Sequences and Series 3. Change 0.4444… to a common fraction. Observe that is an infinite geometric series with   Infinite geometric series sum formula:    

19. A : Sequences and Series 4. How about 7.353535…? (Answer on the next click) Observe that the decimal part is an infinite geometric series with   Infinite geometric series sum formula:       (solution on the next click)  

20. A : Sequences and Series 5. If the 1st and 5th terms of an arithmetic sequence are -5 and 7 respectively, find the sum of the first 21 terms . (2018 MMC Grade 10, ER)         Arithmetic sum formula: But So      

21. A : Sequences and Series 6 . An infinite geometric series has a sum of 12 has first term 8. Find the first term that is less than 1. (2018 MMC Grade 10, ER) An infinite geometric series has its formula for the sum: But             The n th term of a geometric series is defined by We’re given that           From the first five terms, we can see the first term less than 1, that is  

22. A : Sequences and Series 7. Another skill in mathematics is to not find every piece of information for the problem, that you can arrive to the answer without finding everything. For example, in this problem from 2019 MMC Grade 10 ER, one can achieve the final answer without finding the 1 st , 2 nd , or other terms except for the needed one. The answer is on the next click. Answer: 15 (solution on next click) In a geometric sequence, the n th term is defined by therefore         If their product is 3375, then       Taking the cube roots of both sides gives us   But   Therefore (the 9 th term is 15)  

23. Try these!* A : Sequences and Series 8. If the first two terms of an arithmetic progression are 3 and 7, respectively. Find the 10 th term. 9. The first and the sixth terms of a geometric progression are 4 and , respectively. Find the 4 th term. 10. In a sequence defined below, find   *From 2018 MMC Grade 10 ER

24. B: Mathematical Induction -- a method of proof used to prove sequence formula The axiom of mathematical induction -- if a statement involving the natural number has two properties that The statement is true for The statement is true for ; whenever it is true for Then the statement is true for all natural numbers  

25. B: Mathematical Induction Prove   a.)   b.1.)   b.2.)     Using b.1.) then add      

26. B: Mathematical Induction 2. Prove   a.)   b.1.)   b.2.)     Using b.1.) then add        

27. B: Mathematical Induction 3. Prove   a.)   b.1.)   b.2.)     Using b.1.) then add    

28. B: Mathematical Induction 4. Prove   a.)   b.1.)   Multiply both sides by      

29. Try this! B: Mathematical Induction 5. Prove  

30. C: Binomial Theorem Note: The notation can be used interchangeably with the notation since they both mean the same thing.  

31. C: Binomial Theorem  

32. C: Binomial Theorem Factorial -- for every positive whole number ( ) We can also define To prove that Let  

33. C: Binomial Theorem Combination -- For all elements choosing objects at a time Pascal’s Combinatorial Identity  

34. C: Binomial Theorem 1. Expand    

35. C: Binomial Theorem 2. Expand   Since   We can substitute      

36. C: Binomial Theorem 3.1. Find the 5 th term of the expansion of   You can expand the expression by your own, but even using the Pascal’s triangle, it can get tedious. Note that the p th term of an expansion is Where   So:       3.2. Find the 8 th term of the expansion      

37. C: Binomial Theorem 4. Find the third term of the expansion           5. Find the third term of the expansion          

38. C: Binomial Theorem Proof of the Binomial Theorem using Mathematical Induction We are going to prove that   We will use the notation for combination. My laptop is lagging, so yah, I have to write short- handedly .   For     For    

39. C: Binomial Theorem For , we know that     We have predefined in the previous slide     Multiply by distribution. We will omit the left hand side for now     Combine like terms  

40. C: Binomial Theorem   Using the Pascal Combinatorial Identity Taking into account that     This holds true when in the binomial theorem    

41. C: Binomial Theorem 6. Find the coefficient of in the expansion   We can rewrite this as   We have to think two exponents and such that -- their sum is the total power of the expansion (in this case, 9) -- their sum, when included with the product of both the terms in the expression, is the required exponent (in this case, 5) For convenience, we will always let be for the first term, and for the second term In symbols:   The reason for the part is the presence of an exponent of -1. The systems of linear equations for and must be The sum which equates to the expansion power The sum when r and s is multiplied to the powers of their respective variables   We will obtain values of   When solving problems which seeks for the coefficient of a term in the expansion, we will follow the expression Where: is the total power is the first term is the second term        

42. C: Binomial Theorem 7. Find the constant term in the expansion of   The constant term is the term containing the variable with the power of zero. Therefore, we can have and such that   Then we will obtain the values of by system of linear equations     *formula adjusted from the previous slide to avoid confusion with the variables in this problem      

43. C: Binomial Theorem 8. In the expansion of , the numerical coefficient of the 5 th term is? (ECE April 1998)*   *This ECE is very outdated. Do not take this example as a glimpse of what an actual Electronics Engineering Exam will look like nowadays.        

44. C: Binomial Theorem 9. The term involving in the expansion of is? (CE Board )*   *Very outdated Civil Engineering Board Exam We can write this as Then            

45. C: Binomial Theorem 10. Find the 6 th term in the expansion . (CE November 1996)*   *Very outdated Civil Engineering Board Exam        

46. C: Binomial Theorem 11. The constant term in the expansion of is? ( Gillesania , 2004, p.69)   Rewrite:   Then            

47. Try these! C: Binomial Theorem 12. Find the constant term in the expansion of 13. Find the coefficient of in the expansion of