9 2 Arithmetic Sequences and Series An introduction

9. 2 – Arithmetic Sequences and Series

An introduction………… Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term Arithmetic Series Geometric Series Sum of Terms

Find the next four terms of – 9, -2, 5, … Arithmetic Sequence 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms…… 12, 19, 26, 33

Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2 x, 3 x, … Arithmetic Sequence, d = x 4 x, 5 x, 6 x, 7 x Find the next four terms of 5 k, -7 k, … Arithmetic Sequence, d = -6 k -13 k, -19 k, -25 k, -32 k

Vocabulary of Sequences (Universal)

Given an arithmetic sequence with x 38 15 NA -3 X = 80

-19 353 ? ? 63 x 6

Try this one: 1. 5 x 16 NA 0. 5

9 633 x NA 24 X = 27

-6 20 29 NA x

Find two arithmetic means between – 4 and 5 -4, ____, 5 -4 5 4 NA x The two arithmetic means are – 1 and 2, since – 4, -1, 2, 5 forms an arithmetic sequence

Find three arithmetic means between 1 and 4 1, ____, 4 1 4 5 NA x The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence

Find n for the series in which 5 y x 440 3 Graph on positive window X = 16

The sum of the first n terms of an infinite sequence is called the nth partial sum.

Example 6. Find the 150 th partial sum of the arithmetic sequence, 5, 16, 27, 38, 49, …

Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?

Example 8. A small business sells $10, 000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation. So the total sales for the first 2 o years is $1, 625, 000

9. 3 – Geometric Sequences and Series

Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term Arithmetic Series Geometric Series Sum of Terms

Vocabulary of Sequences (Universal)

Find the next three terms of 2, 3, 9/2, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic

1/2 x 9 NA 2/3

Find two geometric means between – 2 and 54 -2, ____, 54 -2 54 4 NA x The two geometric means are 6 and -18, since – 2, 6, -18, 54 forms an geometric sequence

x 9 NA

x 5 NA

*** Insert one geometric mean between ¼ and 4*** denotes trick question 1/4 3 NA

1/2 7 x

Section 12. 3 – Infinite Series

1, 4, 7, 10, 13, …. Infinite Arithmetic 3, 7, 11, …, 51 Finite Arithmetic 1, 2, 4, …, 64 Finite Geometric 1, 2, 4, 8, … Infinite Geometric r>1 r < -1 Infinite Geometric -1 < r < 1 No Sum

Find the sum, if possible:

Find the sum, if possible:

Find the sum, if possible:

Find the sum, if possible:

Find the sum, if possible:

The Bouncing Ball Problem – Version A A ball is dropped from a height of 50 feet. It rebounds 4/5 of it’s height, and continues this pattern until it stops. How far does the ball travel? 50 40 40 32 32 32/5

The Bouncing Ball Problem – Version B A ball is thrown 100 feet into the air. It rebounds 3/4 of it’s height, and continues this pattern until it stops. How far does the ball travel? 100 75 75 225/4

Sigma Notation

UPPER BOUND (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE) LOWER BOUND (NUMBER)

Rewrite using sigma notation: 3 + 6 + 9 + 12 Arithmetic, d= 3

Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r = ½

Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4 Not Arithmetic, Not Geometric 19 + 18 + 16 + 12 + 4 -1 -2 -4 -8

Rewrite the following using sigma notation: Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR: DENOMINATOR: SIGMA NOTATION: