# Geometric Sequences and Series

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Geometric Sequences and Series

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

Geometric Sequences (Type 2)
In geometric sequences, you multiply by a common ratio (r) each time. 1, 2, 4, 8, 16, ... multiply by 2 27, 9, 3, 1, 1/3, ... Divide by 3 which means multiply by 1/3 ie

The nth term of an geometric sequence is denoted by the formula
Where a is the 1st term and r is the common ratio The sum of the first n terms of a geometric series is found by using: Note if r>1 then we can use the formula Which is more convenient

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

Example 2 Find u10 for the geometric sequence 144, 108, 81, 60¾, … Answer a = 144 and r = u10 = arn-1 = 144 (¾)9 = …

Example 3 Find S19 for the geometric sequence … Answer a = 3 and r =

-3, ____, ____, ____

x 9 NA

x 5 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

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