Lesson 2 6 Geometric Sequences By Daniel Christie

Lesson 2. 6 Geometric 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: u 2/u 1 = u 3/u 2 = u 4/u 3 = 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 u 2 = u 1 x r u 3 = u 2 x r = u 1 r 2 u 4 = u 3 x r = u 1 r 3 and in general… un = u 1 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 years. Un = n * rn Special case for problems with years SEE BELOW Example: u 1 = 45000 x 0. 8 = 36000. : u 2 = 45000 x 0. 82 = 28800 u 6 = 45000 x 0. 86 = 11, 796

Geometric Series: Sum of the Terms in a Geometric Sequence Sn = the sum of the geometric Equations: sequence Sn = u 1 (rn - 1 ) n = what power in series = 5 th r-1 u 1 = 1 st term in series = 2 Sn = u 1 (1 - rn) rn = ratio to what power = 2 5 r=2 r-1 (2, 4, 8, 16, 32) (1 st , 2 nd , 3 rd , 4 th , 5 th)

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 2 s = 2+22+23+… 263+264 [s-1] 2 s = s-1+264 2 s-s = -1+264 s = 264 -1 s = 1. 84 x 1019

Thank You Pictures by Daniel Christie