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Lesson 2.6 Geometric Sequences

Lesson 2.6 Geometric Sequences. B y 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]

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Lesson 2.6 Geometric Sequences

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  1. Lesson 2.6Geometric Sequences By Daniel Christie

  2. Homework Page 100-103

  3. 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.

  4. General Term of a Geometric Sequence u2 = u1xr u3 = u2xr = u1r2 u4 = u3xr = u1 r3 and in general… un = u1x rn-1

  5. 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

  6. Geometric Series:Sum of the Terms in a Geometric Sequence • Sn = the sum of the geometric sequence • n = what power in series = 5th • u1 = 1st term in series = 2 • rn = ratio to what power = 25 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)

  7. 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

  8. Thank You Pictures by Daniel Christie

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