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.

Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. The number r is the 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

Ex. 5 The fourth term of a geometric sequence is 125, and the 10th term is 125/64. Find the 14th term. (Assume all terms are positive) a4 = 125 a10 = a4r6 a14 = a10r4 a10 = a14 =

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

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:

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

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