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Chapter 1: Number Patterns 1.6: Geometric Sequences

Chapter 1: Number Patterns 1.6: Geometric Sequences. Essential Question: What is a geometric sequence?. 1.6: Geometric Sequences. Back in section 1.4, we talked about arithmetic sequences. An arithmetic sequence was a sequence that simply added a constant term, d .

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Chapter 1: Number Patterns 1.6: Geometric Sequences

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  1. Chapter 1: Number Patterns1.6: Geometric Sequences Essential Question: What is a geometric sequence?

  2. 1.6: Geometric Sequences • Back in section 1.4, we talked about arithmetic sequences. An arithmetic sequence was a sequence that simply added a constant term, d. • Geometric sequences (a.k.a. geometric progression) are sequences where a common ratio, r, is multiplied to successive terms. • Examples: • {3, 9, 27, 81, …} r = 3 r = ½

  3. 1.6: Geometric Sequences • Recursive Form: • Recursive form for arithmetic sequence: • un = un-1 + d, for n ≥ 2 • RECURSIVE FORM FOR GEOMETRIC SEQUENCE: • un = run-1, for r≠0 and n ≥ 2 • Remember, two things are necessary for a recursive function • Starting point (u1) and the function (un)

  4. 1.6: Geometric Sequences • Explicit Form • If there is a constant number being multiplied over and over, it’s the same as multiplying that common ratio as an exponent • Ex: u2 = u1 ∙ r u3 = u2 ∙ r = (u1 ∙ r) ∙ r = u1 ∙ r2 u4 = u3 ∙ r = (u1 ∙ r2) ∙ r = u1 ∙ r3 • This gives us the explicit form: un = u1 ∙ rn-1

  5. 1.6: Geometric Sequences • Example 4: Explicit • Write the explicit form of a geometric sequence where the first two terms are 2 and -2/5 and find the first five terms of the sequence. • First, we need to find the common ratio, acquired by dividing successive terms: • Explicit Form: un = 2 ∙ (-1/5)n-1 • The sequence begins

  6. 1.6: Geometric Sequences • Example 5: Explicit Form • The 4th term and 9th terms of a geometric sequence are 20 and -640. Find the explicit form. • The first thing we need to do is figure out the common ratio. • The 4th term: u4 = u1rn-1 20=u1r3 • The 9th term: u9 = u1rn-1 -640=u1r8 • Their ratio can be used to find r:

  7. 1.6: Geometric Sequences • Example 5 (Continued) • u4=20, u9=-640 • We now know r=-2 • Find u1 by using u4: • un = u1(-2)n-1 • u4 =u1(-2)3 • 20 =u1(-8) • -5/2=u1 • We have everything we need for our sequence:un = -5/2 ∙ (-2)n-1

  8. 1.6: Geometric Sequences • Partial Sums • The kth partial sum of the geometric sequence {un} with common ratio ≠ 1 is • We can also calculate the partial sums using the sum feature of the calculator: • sum seq(function,x,1,k) where we plug our function in

  9. 1.6: Geometric Sequences • Example 6: Partial Sums • Find the sum: • The first term is -3/2, and the common ratio is ½ • This is the 9th partial sum (9 terms) of the geometric sequence: • Which you can plug into the formula from the last page, or use the calculator: • sum seq((-3/2)*(-1/2)^(x-1),x,1,9) =

  10. 1.6: Geometric Sequences • Exercises • Page 63-64 • 1 – 17 • 23 – 33 • 39 – 41 • odd problems

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