How do I find the sum & terms of geometric sequences and series? Essential Question: What is a sequence and how do I find its terms and sums? How do I find the sum & terms of geometric sequences and series?

Geometric Sequences Geometric Sequence– a sequence whose consecutive terms have a common ratio.

Ex. 1 2, 4, 8, 16, …, formula?, … 12, 36, 108, 324, …, formula?, … Are these geometric? 2, 4, 8, 16, …, formula?, … Yes 2n Yes 4(3)n 12, 36, 108, 324, …, formula?, … No (-1)n /3 1, 4, 9, 16, …, formula? , … No n2

Finding the nth term of a Geometric Sequence an = a1rn – 1

Ex. 2b Write the first five terms of the geometric sequence whose first term is a1 = 9 and r = (1/3).

Ex. 3 Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05 an = a1rn – 1 a15 = (20)(1.05)15 – 1 a15 = 39.599

Ex. 4 Find a formula for the nth term. 5, 15, 45, … an = a1rn – 1 an = 5(3)n – 1 What is the 9th term? an = 5(3)n – 1 a9 = 5(3)8 a9 = 32805

Finite vs. Infinite Geometric Series What was the difference between an infinite sequence and a finite sequence? A finite sequence has a specific number of terms whereas an infinite sequence listing terms continually Finite and infinite series follow the same pattern

sum of a finite geometric series  

Ex. 6 Find the sum of the first 12 terms of the series 4(0.3)n = 4(0.3)1 + 4(0.3)2 + 4(0.3)3 + … + 4(0.3)12 = 1.714

Ex. 7 Find the sum of the first 5 terms of the series 5/3 + 5 + 15 + … r = 5/(5/3) = 3 = 605/3

Ex 8.   2. 3-6+12-24 … n= 5 3. a1= -2, r = 3, Sn= -242

r=3

What would happen if the series was Infinite? Could you find the sum? Arithmetic Infinite: No Sum Geometric Infinite: There can be a sum depending on r value. Geometric – If r > 1, then there is no sum (called Divergent) Geometric – If r< 1 then there is a sum (called Convergent)

Formula for Infinite Convergent Geometric Series:    

Convergent 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 - Divergent Convergent Infinite Geometric -1 < r < 1

Decide if it is convergent or divergent, then Find the sum, if possible:

Decide if it is convergent or divergent, then Find the sum, if possible:

Decide if it is convergent or divergent, then Find the sum, if possible:

Decide if it is convergent or divergent, then Find the sum, if possible:

  2. 125+25+5+1+… 1. 4. Find the common ratio for the infinite geometric series if a1=81, S=60.75   3.

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

upper limit of summation lower limit of summation The sum of the first n terms of a sequence is represented by summation notation. upper limit of summation lower limit of summation index of summation Definition of Summation Notation

Write out a few terms. If the index began at i = 0, you would have to adjust your formula