Knight’s Charge - PowerPoint Presentation

Slide1

Knight’s Charge 8/26/15

Solve: 1) 2) 3)

Review

Have your homework out on your desk (including your triangle).Slide2

TextbooksWrite your name in your textbook in the appropriate place on the inside front cover.Fill out your index card as follows:

Turn in your index card.Remember: These books cost around $108, so TAKE CARE OF THEM. You need to COVER YOUR BOOK!!Student NameGlencoe Precalculus BookBook #:___________Book Condition: NEWSlide3

Check Homework 8/26/15

Set D Practice WkstSlide4

Sequences and SeriesUnit 1Slide5
Slide6

Consider this:Intro to Sequences

A pyramid of logs has 2 logs in the top row, 4 logs in the second row from the top, 6 logs in the third row from the top, and so on, until there are 200 logs in the bottom row. Slide7

A pyramid of logs has 2 logs in the top row, 4 logs in the second row from the top, 6 logs in the third row from the top, and so on, until there are 200 logs in the bottom row. Intro to Sequences

Write and interpret the first 10 terms of the sequence of numbers generated from the example.Identify the pattern in the sequence of numbers.Write the formula for the nth term of the sequence and use it to find the number of logs in, say, the 76th rowCompute the number of logs in the first 12 rows combined.What is the total number of logs in the pyramid? Slide8

Intro to Arithmetic Series:One of the most famous legends in the lore of mathematics concerns German mathematician Carl Friedrich Gauss

.One version of the story has it that in primary school after the young Gauss misbehaved, his teacher, J.G. Büttner, gave him a task: add the numbers from 1 to100. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant, Martin Bartels. Can you?Gauss's realized that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050.Arithmetic SeriesSlide9

Notation Consider the sequence:

 General Sequences the term number (think of it as the term’s place in line) the nth term represents the FIRST term.

represents the SECOND term.

represents the

THIRD term.

represents

the FOURTH term, etc…

the

previous term

the

next term

IMPLICIT FORMULA: requires knowing the previous term

EXPLICIT FORMULA: requires only knowing the desired

n

.

 Slide10

Fill in the chart.General Sequences

SEQUENCEIMPLICIT FORMULAEXPLICIT FORMULA

SEQUENCE

IMPLICIT FORMULA

EXPLICIT FORMULASlide11

Find the first six terms for each sequence:

,

 Slide12

Arithmetic, Geometric, or Neither?An arithmetic sequence

is one where a constant value is added to each term to get the next term. example: {5, 7, 9, 11, …}A geometric sequence is one where a constant value is multiplied by each term to get the next term. example: {5, 10, 20, 40, …}EXAMPLE: Determine whether each of the following sequences is arithmetic, geometric, or neither: a. b. {9, -1, -11, -21, ...} c. {0, 1, 1, 2, 3, 5, 8, 13, 21,...}

GEOMETRIC

ARITHMETIC

NEITHER

Fibonacci

SequenceSlide13

Formal Definition of an Arithmetic SequenceArithmetic Sequences

A sequence is arithmetic if there exists a number d, called the common difference, such that for for .In other words, if we start with a particular first term, and then add the same number successively, we obtain an arithmetic sequence. Slide14

Example: Write an explicit formula for the sequence {10, 15, 20, 25, …}.

Arithmetic SequencesNote: this sequence is arithmetic with a common difference (d) of 5.Make a table of values for the terms of the sequence. Then graph the table. What do you notice about the graph?It’s LINEAR……

Can you write the equation of the line/sequence now?

Yes, the equation of the line is

…

So the formula for the sequence is

.

 Slide15

Example: Write an explicit formula for the sequence {10, 15, 20, 25, …}. Arithmetic Sequences

So how could we write the formula WITHOUT having to graph it? In general, the explicit formula for an arithmetic sequence is given by . Slide16

Example: Fill in the chart for each arithmetic sequence shown.Arithmetic Sequences

SEQUENCEIMPLICIT FORMULAEXPLICIT FORMULA100th term

SEQUENCE

IMPLICIT FORMULA

EXPLICIT FORMULA

100

th

termSlide17

Example: Given

and , find the 100th term of the sequence.  Arithmetic SequencesSlide18

Example: Given

and , find the 25th term of the sequence.  Arithmetic SequencesSlide19

Arithmetic MeansExample: Form an arithmetic sequence that has 3 arithmetic means between 15 and 35.Example: Form an arithmetic sequence that has 4 arithmetic means between 13 and 15.Slide20

Arithmetic SERIESWhat is an arithmetic SERIES? --the SUM

of an indicated number of terms of a sequence. Arithmetic Sequence: Arithmetic Series:  Slide21

Sum of a FINITE Arithmetic SequenceThe sum of a finite arithmetic sequence with common difference d is

.Example: Find the sum of the first 15 terms of the sequence .Example: Find the sum of the first 100 terms of the sequence {-18, -13, -8, -3, 2,…}. Arithmetic SeriesSlide22

Example: Given the sum of the first 20 terms of a sequence that starts with 5 is 220, find the 20th term.

Arithmetic SeriesSlide23

Example: Given the sum of the first 15 terms of an arithmetic sequence is 165 and the first term is , find…

 the common difference.the 15th term.the explicit formula for the sequence.the sum of the first 20 terms of the sequence.Arithmetic SeriesSlide24

Application of Arithmetic SeriesA corner section of a stadium has 14 seats along the front row and adds

one more seat to each successive row. If the top row has 35 seats, how many seats are in that section?Arithmetic SeriesSlide25

HomeworkArithmetic Sequences

Pre-precal review Set J Extra PracticeTextbook p. 605 #1-25 OddSign up to receive texts from me (assignments, extra credit, etc.)2nd period: Text the message honeycutt2 to 810103rd period: Text the message honeycutt3 to 81010