Arithmetic Sequences Objective: To find the next term in a sequence by looking for a pattern To find the nth term of an arithmetic sequence To find the position of a given term in an arithmetic sequence To find arithmetic means Arithmetic Sequences

Precalculus Unit 2 Sequences & Series Do Now: Look at the pattern and write the next number or expression: a) 1000, 500, 250, 125…  b) 1, 2, 4, 7, 11, 16… c) 1, -3, 9, -27… d) 8, 3, -2, -7… e) 2, 2 , 4, 4 … f) 7a + 4b, 6a + 5b, 5a + 6b, 4a + 7b… g) 1, 3, 7, 11, 16… Then determine: “just a sequence” OR Arithmetic OR Geometric We will: To find the next term in a sequence by looking for a pattern To find the nth term of an arithmetic sequence To find the position of a given term in an arithmetic sequence To find arithmetic means

Sequence- a set of numbers {1, 3, 5, 7, …} Vocabulary Arithmetic Sequence- each term after the first is found by adding a constant, called the common difference, d, to the previous term Geometric Sequence – each term after the first is found by MULTIPLYING a constant, called the common ratio, r, to get the next term Sequence- a set of numbers {1, 3, 5, 7, …} Terms- each number in a sequence Common Difference- the number added to find the next term of an arithmetic sequence. Common Ratio - number multiplied to find the next term of an geometric sequence

Arithmetic v. Geometric DO NOW answers Arithmetic v. Geometric 2, 5, 8, 11, 14…. You are ADDING to get the next term Defining nth term: In this case: In words: Each term equals the first term + the difference n-1 times. 3, 6, 12, 24, 48… You are MULTIPLYING to get the next term Defining nth term: R is the common ratio. In this case: In words: Each term equals the first term x the ratio n-1 times.

an-1 Formula for the nth term of an Arithmetic Sequence nth term 1st term # of common the term difference trying to RECURSIVE: A recursion formula for a sequence specifies tn as a function of the preceding term, an-1 EXPLICIT : An explicit formula for a sequence specifies an as a function of n

Formula for the nth term of an Arithmetic Sequence nth term 1st term # of common Find the indicated term. the term difference 3) trying to find

Find the next four terms of each arithmetic sequence

Arithmetic & Geometric SequencEs Arithmetic sequence: each term is formed by adding a constant to the previous term Common difference: the amount being added or subtracted from one term in a sequence to the next Geometric sequence: sequence in which each term is formed by multiplying the previous term by a constant Common ratio: the amount being multiplied T1 or a1: if you have a sequence or series, this is a symbol for the first term Tn or an: this is a general term that represents the nth term. Tn-1: this term represents the term that comes BEFORE

n tn 1 1 2 4 3 7 4 10 5 13 RECURSIVE Equation FOR THE ABOVE ARITHMETIC SEQUENCE WOULD BE: tn = 3(n-1) + 1

What is the is the 22nd term? T22 = 3(22-1) + 1 what would the 11th term be?   T11 = 3(n-1) + 1  What is the is the 22nd term? T22 = 3(22-1) + 1  

Explicit formula: Explicit formula: an equation that explains how to calculate a term in a sequence directly from its first term Explicit equation for the above Arithmetic Sequence : tn = = 3(n) – 2 Calculate the 50th term of this sequence 148 Calculate the 150th term of this sequence 448

Which number term for this sequence will be 46?

Which number term for this sequence will be 46? 16

ANSWERS: Students find equations for the other two problems, and calculate the 50th and 100th term of each sequence… 5) : tn = -1 + 3(n) OR 3(n) -1 6) : tn = -2 – 3(n) OR -3(n) -2

Precalculus Unit 2 Sequences & Series Do Now: take out hw and answer the following: identify the common sum/difference/ratio/divisor, and find the next 3 terms of each problem a. 1, 4, 7, 10, 13…. b. 2, 5, 8, 11, 14…. c. -5, -8, -11, -14… d. 1/2, 1, 2, 4, 8,... e. 1/9, 1/3, 1, 3…. f. -2, 4, -8, 16… CW: DO NOW part 2-VOCAB with ERITREA PARTNER!! PUZZLE TIME.. We will: find the next term in a sequence by looking for a pattern find the nth term of an arithmetic sequence find the nth term of an geometric sequence!! HW: HANDOUT 1-15 (do now) AND STUDY FOR QUIZ TOMORROW!!

Recursive vs. Explicit GEOMETRIC formulaS N Tn 1 1/2 2 1 3 2 4 4 5 8 tn = t1(r)n-1 RECURSIVE formula    What would the Recursive equation be for this sequence? If the 7th term of this equation is 32, what would the 8th term be?

tn = t1(r)n-1 Find the recursion equation for the following 2 sequences 1/9, 1/3, 1, 3…. -2, 4, -8, 16…

Explicit GeoMETRIC formulaS N Tn 1 1/2 2 1 3 2 4 4 5 8 tn = t0(r)n EXPLICIT formula What is the explicit equation for 1/2, 1, 2, 4, 8,...

Explicit GeoMETRIC formulaS N Tn 1 1/2 2 1 3 2 4 4 5 8 tn = t0(r)n EXPLICIT formula     Equation for 1/2, 1, 2, 4, 8,... tn = ½ (2)n Calculate the 50th term of this sequence Calculate the 150th term of this sequence

Using explicit Geometric formula tn = t0(r)n AND in this case: tn = ½ (2)n Which number term for this sequence will be 128? Which number term for this sequence will be 256?

Using explicit Geometric formula tn = t0(r)n Find equations and calculate the 57th and 38th term of each series… 1/9, 1/3, 1, 3….   -2, 4, -8, 16…

Extra Credit There is a new English Language school in Central Square in Cambridge with 26 students. Of each month, 3 new students enroll, in how many weeks will the school have

DO NOW: A sequence contains t1 = 1, t2 = 2 and t5 = 16, Find the EQUATION (recursive AND EXPLICIT) for Geometric sequence Then find t10 for using recursive or explicit formula Find t20 IF DONE-GOLD STAR: Find the Arithmetic formulasequence (d = contains a decimal) reminder;: hw: study + note card

DO NOW: A sequence contains t1 = 1, t2 = 2 and t5 = 16, Find the EQUATION (recursive AND EXPLICIT) for Geometric sequence Then find t10 for using recursive or explicit formula Find t20 Extra Credit : Find the Arithmetic formulasequence (d = contains a decimal) reminder;: hw: study + note card

Complete each statement. 4)170 is the ____th term of –4, 2, 8

Find the indicated term. 5)

Find the missing terms in each sequence

Write an equation for the nth term of the arithmetic sequence

11.2 Arithmetic Series Objective: To find sums of arithmetic series To find specific terms in an arithmetic series To use sigma notation to express sums 11.2 Arithmetic Series

Tn or an: this is a general term that represents the nth term. Arithmetic sequence: each term is formed by adding a constant to the previous term   Common difference: the amount being added or subtracted from one term in a sequence to the next Geometric sequence: sequence in which each term is formed by multiplying the previous term by a constant Common ratio: the amount being multiplied T1 or a1: if you have a sequence or series, this is a symbol for the first term Tn or an: this is a general term that represents the nth term. Tn-1: this term represents the term that comes BEFORE Tn

Vocabulary Arithmetic Sequence- each term after the first is found by adding a constant, called the common difference, d, to the previous term Sequence- a set of numbers {1, 3, 5, 7, …} Terms- each number in a squence Common Difference- the number added to find the next term of an arithmetic sequence. Arithmetic Series- the sum of an arithmetic sequence Series- the sum of the terms of a sequence {1 + 3 + 5 + … +97}

Sum of an arithmetic Series Arithmetic Sequence Arithmetic Series 4, 7, 10, 13, 16 4+7+10+13+16 -10, -4, 2 -10+(-4)+2 Sum of an arithmetic Series Sum of Series # of series 1st term last term

1) Find the sum of the 1st 50 positive even integers.

2) Find the sum of the 1st 40 terms of an arithmetic series in which a1 = 70 and d = -21.

3) A free falling object falls 16 feet in the first second, 48 feet in the 2nd second, 80 feet is the 3rd second, and so on. How many feet would a free-falling object fall in 20 seconds if air resistance is ignored?

4) Find the 1st three terms of an arithmetic series where:

6) Find the sum of each arithmetic series. 5 + 7 + 9 + … + 27

11.3 Geometric Sequences Objective: To find the nth term of a geometric sequence To find the position of a given term in a geometric sequence To find geometric means 11.3 Geometric Sequences

Geometric Sequence: multiplying each term by a common ratio (r). Example: 3, 12, 48,… r = 4 Example: 100, 50, 25… r = ½ Formula: nth term 1st term common ratio # of terms

Find the next 3 terms of each geometric sequence. 1) 2)

Find the 1st four terms of each geometric sequence

Find the nth term of each sequences. 5) 6)

Find geometric means. 7) 4, ____, ____, ____, 324 8) ____, ____, 12, ____, ____, 96

11.4 Geometric Series Objective: To find sums of geometric series To find the specific terms in a geometric series To use sigma notation to express sums 11.4 Geometric Series

Geometric Sum Formula for Series Sum of the nth terms 1st term common ratio nth term Geometric Sequence Geometric Series 1, 3, 9, 27, 81 1 + 3 + 9 + 27 + 81 5, -10, 20, 5 + (-10) + 20

Find the sum of each geometric series. 1) 7 + 21 + 63 + …, n = 10 2) 2401 – 343 + 49 – …, n = 5

Find the sum of each geometric series. 3) 4)

Find for each geometric series described. 5)

Find for each geometric series described. 6)

last term # 1st term # 1st term common ratio # of term Sigma Notation: last term # 1st term # 1st term common ratio # of term Example: 5 + 10 + 20 + 40 + … to 7 terms

Express each series in sigma notation and find the sum 7) 75 + 15 + 3 + … to 10 terms

Express each series in sigma notation. 8)

11.5 Infinite Geometric Series Objective: To find the sums of infinite geometric series 11.5 Infinite Geometric Series

Sum of an Infinite Geometric Series Sum 1st term common ratio

Find the sum of each infinite geometric series, if it exists

2). An old grandfather clock is broken 2) An old grandfather clock is broken. When the pendulum is swung it follows a swing pattern of 25 cm, 20 cm, 16 cm, and so on until it comes to rest. What is the total distance the pendulum swings before coming to rest?

Evaluate. 6)

Evaluate. 7)

11.6 Recursion and Special Sequences Objectives: To recognize and use special sequences To iterate functions 11.6 Recursion and Special Sequences

Vocabulary Recursive Formula- 2 parts: 1st the value of the first term, 2nd shows how to find each term from the term before it Example (Fibonacci Sequence): 1, 1, 2, 3, 5, 8, …

Sequence Sequence Type 9, 13, 17, … arithmetic 7, 21, 63, … geometric 1, 1, 2, 3, 5, 8, … Fibonacci

The Binomial Theorem Objective: To expand powers of binomials by using Pascal’s triangle To find specific terms of binomial expansions

Evaluate by hand and using a calculator. 4)

Sum of an Infinite Geometric Series Items on the Test Arithmetic Sequence Common Difference Arithmetic Series Geometric Sequence Common Ratio Geometric Series Sigma Notation Sum of an Infinite Geometric Series Iterates Binomial Theorem