10.2 – Arithmetic Sequences and Series

An introduction … describe the pattern Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms 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, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k

Vocabulary of Sequences (Universal)

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

?? x 6 353

Try this one: x NA 0.5

9 x 633 NA 24 X = 27

NA x

Find two arithmetic means between –4 and 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, ____, ____, ____, 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 x y X = 16 Graph on positive window

10.3 – Geometric Sequences and Series

Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms 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 To find r, divide any term in the sequence by its preceding term. a 2 /a 1 a 3 /a 2

1/2 x 9 NA 2/3

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

x 9 NA

x 5

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

Find the sum, if possible: What? If possible? What are they talking about? If r is between -1 and 1!

Find the sum, if possible:

Converting repeating decimals to fractions Write the repeating decimal … as a fraction. Write the repeating decimal … as a fraction.

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? / /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? / /4

Sigma Notation Section 12-5

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

Rewrite using sigma notation: Arithmetic, d= 3

Rewrite using sigma notation: Geometric, r = ½

Rewrite using sigma notation: Not Arithmetic, Not Geometric

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