Mathematics (Pg: All )

1. Mathematics (Pg: All ) By:ShaDe’ Phoenix

2. Are you ready to have your brain filled with knowledge? • Take notes; this course gets really complex and you will be required to answer questions after every section. I will choose at random. • Please ask billions of questions if you do not understand something.

3. Let’s start with the basics (Pg: 5-22) • Multiplication is an easier form for repeated addition Instead of 3+3+3+3 you could say 3*4

4. The Multiplication Principle • When listing the possibilities for k items, the total number of entries in this list is given by n sub 1*n sub 2* n sub 3….*n sub k Ex: in this definition, n sub 3 would be the number of possibilities for the third item (Refer to example 1.1e)

5. Permutations • An arrangement of objects from a group where NO object can be used more than once and the ORDER of selection MATTERS. • Keep in mind with and without replacement without replacement: factorials with replacement: exponents Permutation formula: n!/(n-k)! k being the number of objects in the list that you want; n-k is you canceling out all the other objects

6. Factorials • The symbol ! read aloud (pertaining to mathematics) means factorials • Factorials are the product of all whole numbers starting from the number indicated down to one Ex: 5! is 5*4*3*2*1 which equals 120

7. Combinations • An arrangement of objects from a group where NO object can be used more than once and ORDER of selection DOES NOT matter Combinations formula: (n!/(n-k)!)/k!

8. Notation and Calculator input • Notation: (n over k) which is read n choose k • Calculator input type your n click math arrow to your right to the fourth column, select either nPr (for permutation) or nCr (for combination) *(permutations only work for without replacement problems)* type your k press enter

9. Algebra (Pg: 23-77)

10. Again, we'll start off with something basic

11. Sequences • Sequence- a list of objects presented in a particular order -The objects in a sequence are called the terms of the sequence (usually denoted by the variables a or x) -The terms of sequences aren't always numbers - they can be shapes or even words -The position of a term in a sequence is called the index (usually denoted as i or k) of the term (written as the subscript of the term) -Usually, the relationship between the index and the term of the sequence becomes apparent Based on the pattern 2,4,6,8,10 what would you guess the next 3 numbers in this sequence would be? What about the next 1 in this sequence? Monday Tuesday Wednesday • Nice sequences: can be extended in a logical manner

12. Mathematically writing sequences • { } tell you that what you are about to solve is a sequence not an equation Ex: {x sub i=2i} when i is 1 the term (or x) is 1*2 or 2 and so on • Now you try. Give me 4 terms of the sequence {x sub i=i^3} -The name of this sequence is the sequence of cubed numbers • Now give me the sequence that will generate these numbers 3,6,9,12 • What about 3,7,11,15,19?

13. Finite sequences • Finite sequences do not go on forever … so how do you limit a sequence?

14. Arithmetic and Geometric Sequences

15. Recursive formula • (in terms of a sequence) is a formula that declares the starting value or values for the sequence and how the other terms (subsequent terms) are made from the previous terms. -The formula alone only shows how to get from one term to the next and thus can not describe the sequence alone • The recursive formula for the sequence 3,6,12,24,48 would be x sub 1=3; x sub i=x sub x-I * 2 (direct your attention to the board for clarity)

16. Direct formulas • Generate a sequence directly -Like x sub i=4i-1 . If you plug in any number for the index you can find the term -whereas with a recursive formula you need the previous term in order to find the next term

17. Want to try? • What are the terms of the sequence given by the recursive formula: A sub 1=1 A sub 2=1; A sub i= a sub i-1+ a sub i-2 ? • First ask yourself, “What is this problem saying?”

18. Arithmetic Sequence • A sequence where a fixed amount is added to move from one term to the next -there is a constant difference between consecutive terms -The constant difference can also be negative (it will still remain arithmetic) • The constant difference of an arithmetic sequence is usually represented by the variable d • The first term in the sequence is usually represented by the variable a • Recursive arithmetic formula: x sub i=a; x sub i=x sub i-1 + d • Direct formula: x sub k= a+(k-1)*d

19. Now apply it • What are the next 3 terms in the arithmetic sequence 41,38,35,32? • What is the 201st term in the arithmetic sequence 41,38,35,32? -Think: what key words tell you which formula to use? • What is the first term of the sequence with x sub 54= 136 and x sub 77=205?

20. Geometric Sequences • A sequence with a constant ratio (usually represented by the variable r) between consecutive terms -This tells you that you will now being using multiplication or division -The first term for a geometric sequence is usually represented by the variable a • What is the ratio for the geometrical sequence 36,12,4,4/3? -Think: how are you getting from one term to the next? • Recursive formula: x sub 1=a; x sub i=r*x sub i-1 • Direct formula: x sub k= a*r^k-1

21. Try it out for size • What are the next 3 terms for the geometric sequence 2,4,8,16,32? • What is the 20th term for the geometric sequence 2,4,8,16,32? • What are the first 6 terms of a geometric sequence when the 2nd term is 2 and the 6th term is 8?

22. Series • The sum of the terms in a sequence

23. Solving Arithmetic Series • What is 1+2+3…+97+98+99+100? Think: “Work smarter, not harder”

24. Notice that if you form pairs they all have a sum of 101 (1+100, 2+99, 3+98, etc.) and there are 50 pairs (100/2). So, the sum can be found by multiplying the common sum by the number of pairs (101*50). • However, if there is not a common sum among the pair, this process will not work

25. Solving Arithmetic Series with an Odd Number of terms • Ways to do so: Add an extra term to the sequence (giving it an even number of terms) and then subtract the term you added at the end Omit a term and then add it to the end Try to figure out the middle term that has no pair • Given this knowledge, how would you go about solving the arithmetic series 41+38+35+…+(-1)+(-4)+(-7) Hint: This series has a constant difference of -3 and 17 terms

26. You could solve that problem by adding a term (-10) to make 9 pairs and multiplying that but the common sum (41+-10=31, 31*9=279) then subtracting the term you added (279- -10 or 279+10) • But, there is a simpler way

27. Formula • DIRECT YOUR ATTENTION TO THE BOARD

28. Given what you have learned… • Solve: “An arithmetic series equals 624. This first term is 3, and the second term is 5. What is the last term in this series?” Hint: Keep in mind the situation

29. Geometric Series • S (sum)= a*r^k-a / r-1 a- the first term r- the constant ratio between consecutive terms k- the number of terms • If you don’t have know the number of terms the equation would be S (sum)= x sub k+1 - x sub 1 / r-1 X sub k+1- the term after the last term in the series (this term is not included in the series) x sub 1- the first term in the series r- the constant ratio between consecutive terms • This method would not work if the ratio was 1 (1-1=0, any number divided by 0 is undefined)

30. Infinite series • If an arithmetic series has an infinite number of terms, the series will equal positive or negative infinity depending on the constant difference (d). If d > 0 the series will equal positive infinity. If d < 0 the series will equal negative infinity.

31. Infinite Geometric Series • An infinite geometric series equals a finite number if x sub k (the “last term”) is approximately equal to 0, which would mean that the ratio would have to be causing the terms in the series to decrease (like r= ½) • Formula: S (sum)= a/1-r a- the first term r- the constant ratio between consecutive terms • Example: The value of the infinite geometric series 2, -4/3, +8/9, -16/27 is 6/5 because: the first term is 2 the ratio is -2/3, meaning the values of each individual term, gets smaller and smaller as the sequence goes on, meaning x sub k is approximately equal to 0

32. Sigma Notation- a lot of board work • Mathematicians use this to write out complicated/ long sums in shorthand

33. Write the following in sigma notation 1+4+9+16+…+196+225+256

34. Formulas • Direct attention to board

35. Polynomials • Definition- an algebraic object consisting of terms Each term is made up of a variable (usually x) which is raised to a non-negative power and has a coefficient. The highest power of x with a non-zero coefficient is called the degree of the polynomial • Knowing this, what is the degree of 3x^3+7x^2+5 ? • Polynomials can still be thought of as a series and written in sigma notation

36. Adding and Subtracting polynomials • Combining like terms • What sequence occurs if the following 2 sequences are added: (2,4,6,8,10)+(5,10,15,20,25) • Subtract the following polynomial: (6+2x-2x^2+-6x^3)-(3x+12x^2+21x^3)

37. Multiplying Polynomials • I am sorry, I don’t like the method in the book. So, I’ll teach the book method and foiling. • What is (x^2+5x-6)*(3x^3-7x^2+4) The book: since there are 3 terms in each quantity you can tell that you will be looking at 9 terms (3*3) The highest power will be x^5 which will result from x^2*3x^3 and generate 3x^5 the coefficient for x^4 will be generated by x^2*-7x^2=-7x^4 and 5x*3x^3=15x^4 the terms that will generate x^3 are -6*3x^3=-18x^3 and 5x*-7x^2=-35x^3 the terms that will generate x^2 are x^2*4=4x^2 and -6*-7x^2=42x^2 The terms that will generate x are 5x*4=20x The terms that will generate the constant term are -6*4=-24 • When combined and simplified, you get the answer of 3x^5+8x^4-58x^3-38^2+20x-24

38. Now try it with either method • (x-2)(x^2*3x*3)(2x*-4) Think: how many terms should you have?

39. The Binomial Expansion Theorem • Definition- taking a binomial to a power • Binomials-polynomials with 2 terms Let’s start simple: 2(5x-3)^2 What about (x+2)^3 ? What about (x+1)^10?

40. That last one would take forever ! • So, instead we will represent it in sigma notation • Try writing (4x+2)^20 in sigma notation

41. Formula • When expanding a binomial that is in the form (x+y)^n realize that each term in this expansion is in the form (n choose k) * x^k *y^n-k Where: x- is one term y- is another term n- the power the binomial is being raised to k-the power the first term is being raised to

42. Apply it • What is the coefficient of z^5 in the expansion (z-2)^10?

43. Compound interest • Simple interest- fee fixed per time period • Compound interest- interest charged on the interest previously charged

44. Try it out! • Mrs. Mal borrows $1800 dollars from the bank to pay for her trip to Moscow. She is charged 5% annual interest but interest is compounded every quarter. Assuming Mrs. Mal does not pay back any of the money along the way, how much money will she owe after 2 years? Think: What is annual interest? How often is quarterly?

45. Formula/Apply it • Honey Boo Boo invests $1,250 in a certificate of deposit (CD) that earns 3.7% interest compounded monthly. When she goes back to the bank in 4.5 years and collects her money, how much will she have earned?

46. Annuities • Megan invests in an annuity that earns 3.6% interest compounded quarterly. If Megan deposits $160 a quarter (and does not withdraw any money from her account), how much money will she have after 15 years?

47. Solution • Only the first $160 deposited is in the account for all 60 quarters, which means it is compounded 59 times; from there, the amount of times each $160 is compounded decreases until the last $160 added is not compounded at all. This is an example of a sum of a geometric series and can be written using that formula.

48. Annuity Formula/Apply it • ShaDe’ invests $1200 a year in a retirement fund that earns 4% annual interest for 43 years. How much money does ShaDe’ have in her fund when she retires? • How much did she invest? How much did she earn?

49. Loans • Samantha Taylor borrows $150,000 at 7% annual interest compounded monthly to buy a house. If Samantha takes out a 20 year mortgage, what will her monthly house payment be? • How much will she end up paying the bank?

50. Now that you know the formula… • Tanairy is ready to buy a house. She feels she can afford a $1000 monthly mortgage. If current mortgage interest rates are 6% and she is interested in a 25 year loan, about how much money can Tanairy expect to be able to borrow?