Lesson 2.6 Geometric Sequences

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Presentation transcript:

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: 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

Geometric Series: Sum of the Terms in a Geometric Sequence Sn = the sum of the geometric sequence Equations: Sn = u1 (rn - 1 ) n = what power in series = 5th r - 1 u1 = 1st term in series = 2 Sn = u1 (1 - rn) rn = ratio to what power = 2 5 r = 2 (2, 4, 8, 16, 32) (1st , 2nd , 3rd , 4th , 5th)

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

Pictures by Daniel Christie Thank You Pictures by Daniel Christie