Problem Solving; Venn Diagrams; Patterns; Pascal’s Triangle; Sequences MATH 2160 1st Exam Review Problem Solving; Venn Diagrams; Patterns; Pascal’s Triangle; Sequences

Problem Solving Polya’s 4 Steps Understand the problem Devise a plan Carry out the plan Look back

Problem Solving Strategies for Problem Solving Make a chart or table Draw a picture or diagram Guess, test, and revise Form an algebraic model Look for a pattern Try a simpler version of the problem Work backward Restate the problem Eliminate impossible situations Use reasoning

Problem Solving How many hand shakes? Playing darts Tetrominos Who am I? Triangle puzzle

Venn Diagrams Vocabulary Intersection Universe Union Element Set Subset Disjoint Mutually Exclusive Finite Intersection Union Compliment Empty Set Infinite

Venn Diagrams What can you say about A and B? A Ç B = Æ A È B = {A, B} A and B are mutually exclusive or disjoint A B

Venn Diagrams What can you say about A and B? A Ç B = A È B = A’ Ç B =

Venn Diagrams What can you say about A, B, and C? A Ç B  C =? A È B  C =? (A Ç C)  B =? A Ç (C  B) =? (A Ç B)  C =? C Ç (A  B) =? (B Ç C)  A =? B Ç (C  A) =? (A’ Ç B)  C =? (A’ È B)  C =? A’ Ç B’  C’ =? A’  B’  C’ =? Etc. B A C

Patterns Triangular Numbers Etc. T1 T2 T3 T4 Tn = Tn-1 + n

Patterns Square Numbers Etc. S1 S2 S3 S4 Sn = n2

Patterns Rectangular Numbers R1 R2 R3 R4 Etc. Rn = n (n + 1)

Pascal’s Triangle Expanding a binomial expression: (a + b)0 = 1 (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

Pascal’s Triangle Vocabulary Expansion – the sum of all of the terms Coefficient – the number in front of the variable(s) for a particular term Variable(s) – the letters AND their exponents for a particular term Term – the coefficient AND the variable(s)

Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

Pascal’s Triangle Magic 11’s 110 1 111 1 1 112 1 2 1 113 1 3 3 1 110 1 111 1 1 112 1 2 1 113 1 3 3 1 114 1 4 6 4 1 Fails to work after this…

Arithmetic Sequences The difference between any two consecutive terms is always the same. Examples: 1, 2, 3, … 1, 3, 5, 7, … 5, 10, 15, 20, … Non-Examples 1, 4, 9, 16, … 2, 6, 12, 20, …

Arithmetic Sequences The nth number in a series: Example an = a1 + (n – 1) d Example Given 2, 5, 8, …; find the 100th term n = 100; a1 = 2; d = 3 a100 = 2 + (100 – 1) 3 a100 = 2 + (99) 3 a100 = 2 + 297 a100 = 299

Arithmetic Sequences Summing or adding up n terms in a sequence: Example: Given 2, 5, 8, …; add the first 50 terms n = 50; a1 = 2; a50 = 2 + (50 – 1) 3 = 149 S50 = (50/2) (2 + 149) S50 = 25 (151) S50 = 3775

Arithmetic Sequences Summing or adding up n terms in a sequence: Example: Given 2, 5, 8, …; add the first 51 terms n = 51; a1 = 2; a2 = 5; a51 = 2 + (51 – 1) 3 = 152 S51 = 2 + ((51-1)/2) (5 + 152) S51 = 2 + (50/2) (5 + 152) S51 = 2 + 25 (157) S51 = 2 + 3925 S51 = 3927

Geometric Sequences The ratio between any two consecutive terms is always the same. Examples: 1, 2, 4, 8, … 1, 3, 9, 27, … 5, 20, 80, 320, … Non-Examples 1, 4, 9, 16, … 2, 6, 12, 20, …

Geometric Sequences The nth number in a series: Example an = a1 r(n-1) Given 5, 20, 80, 320, …; find the 10th term n = 10; a1 = 5; r = 20/5 = 4 a10 = 5 (4(10-1)) a10 = 5 (49) a10 = 5 (262144) a10 = 1310720

Geometric Sequences Summing or adding up n terms in a sequence: Example: Given 5, 20, 80, 320, …; add the first 7 terms n = 7; a1 = 5; r= 20/5 = 4 S7 = 5(1 – 47)/(1 – 4) S7 = 5(1 – 16384)/(– 3) = 5(– 16383)/(– 3) S7 = (– 81915)/(– 3) = (81915)/(3) S7 = 27305

Fibonacci Sequences 1, 1, 2, 3, … Seen in nature Golden ratio Pine cone Sunflower Snails Star fish Golden ratio (n + 1) term / n term of Fibonacci Golden ratio ≈ 1.618

Test Taking Tips Get a good nights rest before the exam Prepare materials for exam in advance (scratch paper, pencil, and calculator) Read questions carefully and ask if you have a question DURING the exam Remember: If you are prepared, you need not fear