 # 13.1, 13.3 Arithmetic and Geometric Sequences and Series

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13.1, 13.3 Arithmetic and Geometric Sequences and Series
13.5 Sums of Infinite Series Objective To identify an arithmetic or geometric sequence and find a formula for its n-th term; 2. To find the sum of the first n terms for arithmetic and geometric series.

ADD To get next term MULTIPLY To get next term
1. Arithmetic Sequence and Series Arithmetic Series Sum of Terms Geometric Series Sum of Terms Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term + d + d + d + d + d + d a1 a2 a3 a4 a5 a6 an - 1 an  r  r  r  r  r  r

The notation of a sequence can be simply denoted as
{an}, meaning a1, a2, …, an, … For any three consecutive terms in an arithmetic sequence, … , ak – 1 , ak , ak + 1 , … ak is called the arithmetic/geometric mean of ak – 1 and ak + 1 if the sequence is arithmetic/geometric.

If three consecutive terms,
… , ak – 1 , ak , ak + 1 , … in an arithmetic sequence, then ak = ak – 1 + d ak = ak + 1 – d ak + ak = ak – 1 + ak + 1 2ak = ak – 1 + ak + 1 Or, ak is the arithmetic average of ak – 1 and ak + 1.

If three consecutive terms,
… , ak – 1 , ak , ak + 1 , … in a geometric sequence, then Or, ak is the geometric average of ak – 1 and ak + 1.

Arithmetic mean ak + ak = ak – 1 + ak + 1 2ak = ak – 1 + ak + 1 Geometric mean

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 7

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 38 15 NA -3 a1 = 80

-19 353 x 63 x 6

Try this one: 1.5 16 x NA 0.5

9 x 633 NA 24 n = 27

-6 29 20 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 n = 16

Find the number of multiples of 4 between 35 and 225.
The difference of any two consecutive multiples of 4 is 4. Those multiples of 4 form an arithmetic sequence. Since 35 < 36 (the first multiple of 4 we are looking for), we can write the nth term of an arithmetic sequence as

ADD To get next term MULTIPLY To get next term
2. Geometric Sequences and Series Arithmetic Series Sum of Terms Geometric Series Sum of Terms Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term + d + d + d + d + d + d a1 a2 a3 a4 a5 a6 an - 1 an  r  r  r  r  r  r

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

-3, ____, ____, ____

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

13.5 Infinite Geometric 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 or r  -1 No Sum |r|  1 Infinite Geometric -1 < r < 1 |r| < 1

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 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? 100 100 75 75 225/4 225/4

An old grandfather clock is broken
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? 25 25 20 20 16 16

Find what values of x does the infinite series converge?
This infinite geometric series with r = 2x/7. By the theorem of convergence for the infinite geometric series, the series converges when | r | < 1. Or, or

Find what values of x does the infinite series converge?
This infinite geometric series with r = 3(x – 1)/2. By the theorem of convergence for the infinite geometric series, the series converges when | r | < 1. Or, or So

Challenge Question Sum of non-arithmetic and non-geometric series Find the formula for the sum of the first n terms of the series, or Sn, then find the sum The integer part of the each term forms a arithmetic series and the fraction part of each term forms a geometric series: We apply two formula, one for arithmetic series and one for geometric series to get the overall formula for Sn:

Challenge Question Textbook P. 503 #30 (Sum of non-arithmetic and non-geometric series.) Find the formula for the sum of the first n terms of the series, or Sn, then find the sum of the infinite series by using limit. Each term can be decomposed to half of the difference of two fractions – numerators are all 1 and the denominators are the two integers, respectively: Starting from the second group, each first term within each parentheses can be cancel out by the second terms in the preceding group, respectively.

The only terms are not canceled out are 1 and –1/(2n + 1). So,
Therefore the limit of the sum is

Challenge Question Given an arithmetic sequence {an}, how can you make a geometric sequence? We make a geometric sequence as Since It shows that the new built sequence is a geometric sequence of common ratio r = ed .

Challenge Question Given a geometric sequence {an}, how can you make an arithmetic sequence? We make an arithmetic sequence as Since It shows that the new built sequence is an arithmetic sequence of common difference d = ln|r| .

Index must be the same variable
Sigma Notation UPPER BOUND (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE) LOWER BOUND (NUMBER) Index must be the same variable

The relationship between Sn , Sn – 1 and an

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

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

CAUTION CAUTION

Some Properties of Sigma Notation
1. Ex. 2. Ex. 8 times 3. Ex.

Some Useful Series 1. 2. 3. 4. 5.

Assignment 13.1 P #4 – 36 (x4), 17 – 41(odd), 44 – 45 34, 38, 47, 49, 51. 13.3 P #2 – 12 (even), 11, 15, 18 – 21, 23, 27, 30 13.5 P #2 – 18 (even), 21 – 35 (odd)