Lesson 2.6Geometric Sequences

By Daniel Christie

Homework

Page 100-103

Explained: Geometric Sequences

A sequence is geometric if the quotient between a term in the sequence and it’s previous term is a constant [usually called a common ratio]

Example: u2/u1 = u3/u2 = u4/u3 = r

Or: 2/1 = 4/2 = 8/4 = 16/8 = 2

Explanation: The common ratio is 2 because every fraction in the set equals 2.

General Term of a Geometric Sequence

u2 = u1 x r

u3 = u2 x r = u1r2

u4 = u3 x r = u1 r3

and in general…

un = u1 x rn-1

Application of Geometric Sequences

Problem: A car costs $45000. It loses 20% value every year. How much is the car worth in 6 years?

Un is the number of years. 0.8 is the common ratio.

Un = n * rn-1

Special case for problems with yearsUn = n * rn

Special case for problems with years SEE BELOW

Example: u1 = 45000 x 0.8 = 36000 .: u2 = 45000 x 0.82 = 28800

u6 = 45000 x 0.86 = 11,796

Sn = the sum of the geometric sequence

n = what power in series = 5th

u1 = 1st term in series = 2

rn = ratio to what power = 2

5

r = 2

Equations:

Sn = u1 (rn - 1 )

r - 1

Sn = u1 (1 - rn)

r - 1

(2, 4, 8, 16, 32)

(1st , 2nd , 3rd , 4th , 5th)

Geometric Series:Sum of the Terms in a Geometric Sequence

Geometric Series:Sum of the First n Terms of a Geometric Sequence

An example geometric series: 1,2,4,8,16,32,64,128

Example: 1+2+22+23+24+25…+263

2s = 2+22+23+… 263+264 [s-1]

2s = s-1+264

2s-s = -1+264

s = 264-1

s = 1.84 x 1019

Thank You

Pictures by Daniel Christie