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

Universe Element Set Subset Disjoint Mutually Exclusive Finite

Intersection Union Compliment Empty Set Infinite

What can you say about A and B?

A B = A B = {A, B} A and B are mutually exclusive or disjoint

Venn Diagrams

A B

Venn Diagrams What can you say about A and B?

A B = A B = A’ B = A’ B = A B’ = A

B’ = A’ B’ = A’ B’ =

BA

Venn Diagrams What can you say

about A, B, and C? A B C =? A B C =? (A C) B =? A (C B) =? (A B) C =? C (A B) =? (B C) A =? B (C A) =? (A’ B) C =? (A’ B) C =? A’ B’ C’ =? A’ B’ C’ =?

Etc.

BA

C

Patterns Triangular Numbers

Etc.

T1 T2 T3 T4

Tn = Tn-1 + n

n

1xn xT

2

)1n(nTn

Patterns

Square Numbers

Etc.

S1 S2 S3 S4

Sn = n2

Patterns

Rectangular Numbers

Etc.

R1 R2 R3 R4

Rn = n (n + 1)

Rn = n2 + n

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

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

Pascal’s Triangle

Magic 11’s110 1111 1 1112 1 2 1113 1 3 3 1114 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: 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

n1n aa2

nS

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

n21n aa2

1-naS

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: an = a1 r(n-1)

Example 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

r1

r1aS

n1

n

Fibonacci Sequences

1, 1, 2, 3, … Seen in nature

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