Sequences …and Series

Exploring Arithmetic and Geometric Sequences

Sequences 1, 2, 3, 4, 5, 6, 7, … 1, 3, 5, 7, 9, 11, … 2, 4, 8, 16, 32, 64, …1, 1, 2, 3, 5, 8, 13, 21, …27, 20, 13, 6, -1, -8, …40, 20, 10, 5, 2.5, 1.25, …3, 9, 27, 81, 243, 729, …

Arithmetic Sequences Ex.1) 2, 5, 8, 11, 14, 17, 20, …

Ex. 2) 4, 9, 14, 19, 24, 29, 34, …

Ex. 3) 1, 8, 15, 22, 29, 36, …

Notes: Lists of Numbers Pattern Set difference between terms

Generalizing Arithmetic Sequences Ex.1) 2, 5, 8, 11, 14, 17, 20, …

5 – 2 = 3, 8 – 5 = 3, 11 – 8 = 3 … 20 – 17 = 3 Difference = 3 = d (2 + 3) = u2 , (5 + 3) = u3 , (u2 + 3) = u3 (2 + 3 + 3) = u3 , 2 + 2(3) = u3 , 2 = 3 – 1 un = (2 + 3(n – 1))

Ex. 3) 1, 8, 15, 22, 29, 36, … 8 – 1 = 7, 15 – 8 = 7, 22 – 15 = 7 … 36 – 29 = 7 d = 7 (1 + 7) = u2 , (8 + 7) = u3 (1 + 7 + 7) = u3 , 1 + 2(7) = u3 , 2 = 3 – 1 un = (1 + 7(n – 1))

Notes: Determine common difference General formula: un = u1 + (n – 1)d

Geometric Sequences Ex. 1) 2, 4, 8, 16, 32, 64, …

Ex. 2) 1000,100, 10, 1, 0.1, 0.01, …

Ex.3) 3, 9, 27, 81, 243, 729, …

Notes: List of numbers Pattern [Multiplication] Common ratio [r]

Generalizing Geometric Sequences Ex. 1) 2, 4, 8, 16, 32, 64, …

4÷2 = 2, 8÷4 = 2, … 64÷32 = 2 Common ratio r = 2 u1×2 = u2 , u2 ×2 = u3 , (u1×2)×2 = u3 u3 = u1×(22), thus un = u1(2n-1)

Ex. 2) 1000,100, 10, 1, 0.1, 0.01, … 100÷1000 = 1/10 , 10÷100 = 1/10 Common ratio r =1/10 1/10× u1 = u2 ,1/10× u2 = u3, 1/10× (1/10× u1 )= u3 u3 = u1×(1/10)2

, thus un = u1(1/10)n-1

Notes: Find common ration r [dividing one term by previous term] General formula: un = u1(rn-1)

Series Arithmetic Series

2 + 5 + 8 + 11+ 14 +17 + 20 +… 4 + 9 +14 +19 + 24 + 29 + 34 +…

Geometric Series 2 + 4 + 8 + 16 + 32 + 64… 3 + 9 + 27 + 81 + 243 + 729…

What is a series?Lists of numbers with patternsTerms of sequences added together

Applications What are Arithmetic Sequences used for?

Solving Math Problems Used in conjunction with Series [Arithmetic Series]

Can be used to sum up long lists of numbers Can be used to sum up prices Ex. Theater tickets are sold for $60 for odd numbered tickets

and $40 for even numbered tickets. There are 50 seats in the theater, how much money do even tickets bring in if all are sold?

What are Geometric Sequences used for? Solving Compound Interest

When money is put into a bank it gains a percentage interest. At any given point in time geometric series can determine

how much money is in the bank, with interest. To calculate population growth, often in biology.

Recognizing Sequences 1, 2, 3, 4, 5, 6, 7, … 1, 3, 5, 7, 9, 11, … 2, 4, 8, 16, 32, 64, …1, 1, 2, 3, 5, 8, 13, 21, …27, 20, 13, 6, -1, -8, …40, 20, 10, 5, 2.5, 1.25, …3, 9, 27, 81, 243, 729, …

Properties

Arithmetic Sequences Geometric Sequences A list of numbers There is a pattern and order Get one term by adding a

number to the previous term in the sequence

There is a common difference d

n represents position in the list (nth term)

General formula: un=u1+(n-1)d Series adds each term in a

sequence together Can be used to sum lists of

numbers, cost and money

A list of numbers There is a pattern and order Get one term by multiplying

the previous term by a set number

There is a common ratio r n represents position in the

list (nth term) General formula: un = u1rn-1 Series adds each term n a

sequence together Can be used to determine

growth, population growth and interest

Properties - Similarities

Arithmetic Sequences Geometric Sequences A list of numbers There is a pattern and

order Get one term by adding a number to

the previous term in the sequence There is a common difference d n represents position

in the list (nth term) General formula: un=u1+(n-1)d Series adds each term

in a sequence together Can be used to sum lists of numbers,

cost and money

A list of numbers There is a pattern and

order Get one term by multiplying the

previous term by a set number There is a common ratio r n represents position

in the list (nth term) General formula: un = u1rn-1 Series adds each term

n a sequence together Can be used to determine growth,

population growth and interest

Properties - Differences

Arithmetic Sequences Geometric Sequences A list of numbers There is a pattern and order Get one term by adding a

number to the previous term in the sequence

A common difference d n represents position in the list (nth term) Gen. formula: un=u1+(n-

1)d Series adds each term in a sequence

together Can be used to sum lists of

numbers, cost and money

A list of numbers There is a pattern and order Get one term by

multiplying the previous term by a set number

There is a common ratio r n represents position in the list (nth

term) General formula: un = u1rn-1 Series adds each term n a sequence

together Can be used to determine

growth, population growth and interest

Compare and Contrast

Lists of numbers

Pattern and order

n is the position in the sequence

series adds the terms in the sequence together

•has common difference d

•get one term by adding or subtracting a set number to the previous term

•general formula: un = u1 + (n – 1)d

•Can be used to find sums of long lists of numbers

Has a common ratio r

Get one term by multiplying the previous term by a set numberGeneral formula:

un = u1rn-1

Can be used to figure out compound interest and population growth

Now for Review Get into groups of four

Review information presented in class

Write helpful information on index cards [keep it on the lines – one side]

5 minutes