Math 20-1 Chapter 1 Sequences and Series Teacher Notes 1.3 Geometric Sequences

1.3 Geometric Sequences Math 20-1 Chapter 1 Sequences and Series Many types of sequences can be found in nature. The Fibonacci sequence, frequently found in flowers, seeds, and trees, is one example. A geometric sequence can be approximated by the orb web of the common garden spider. The lengths of the sections of the silk between the radii for this section of the spiral produce a geometric sequence. 8, 12, 18, 27, 40.5, 60.75 http://www.youtube.com/watch?v=LQ5BSt85Qt4&feature=youtube_gdata_player 1.3.1

Geometric Sequences Examples Non-Examples http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=139&ClassID=135423 Geometric Sequences Examples Non-Examples 2, 10, 50 … 250 10, 15, 20, 25, … -4, 12, -36, 108, … -324 5, 4, 3, 2, 1, …, -102, -103 Where would these sequences go? Geometric Sequences What do you think makes a sequence geometric? What is the next term in each geometric sequence? 1.3.2

Characteristics of a Geometric Sequence 8, 12, 18, 27, 40.5, 60.75 Is the relationship between consecutive terms of the geometric sequence different from an arithmetic sequence? For several pairs of consecutive terms in the sequence, divide a term by the preceding term. What did you notice? An ordered list of terms, separated by commas, in which the ratio of consecutive terms is constant is called a geometric sequence. Multiplying any term by a constant called the common ratio, r, will give the next term in the geometric sequence. 1.3.3

Consider the sequence a) What is the value of t1? 4 t4? 32 4, 8, 16, … a) What is the value of t1? 4 t4? 32 b) Determine the value of the common ratio. c) What strategies could you use to determine the value of t10? d) What would the graph of the geometric sequence differ from the graph of an arithmetic sequence? 1.3.4

Deriving a Rule for the General Term of a Geometric Sequence Terms Sequence Sequence Expressed using first term and common ratio General Sequence 4 8 16 32 4 4(2) 4(2)(2) 4(2)(2)(2) 4 (2)…(2) A geometric sequence is a sequence that has a constant common ratio, r, between successive terms. Position of term in the sequence tn = t1 rn-1 General term or nth term First term Common ratio What assumptions are made? 1.3.5

tn = t1 rn-1 tn = t1 rn-1 tn = 4(2)n-1 t10 = 4 (2)10-1 = 4 (2)9 4, 8, 16 … parameters t1 and r must be defined Determine the value of t10. Write the expression for the general term. tn = t1 rn-1 Explicit Definition t1 = 4 n = var r = 2 tn = t1 rn-1 t1 = 4 n = 10 r = 2 t10 = ? tn = 4(2)n-1 t10 = 4 (2)10-1 = 4 (2)9 t10 = 2048 Is it mathematically correct to multiply the 4 and 2? t10 = 4(2)10-1 t10 = 4(2)9 t10 = 2048 1.3.6

tn = t1 rn-1 tn = 1(2)n-1 tn = (2)n-1 t9 = (2)9-1 t9 = 256 t1 = 1 When Spider-Man was bitten, the radioactive spider injected 1 mg of venom into his body. The venom concentration doubled every hour. Write an expression that models the concentration of venom in Spider-Man’s body over time. tn = t1 rn-1 t1 = 1 r = 2 n = var tn = 1(2)n-1 How many mg were in Spider-Man’s blood stream eight hours after being bitten? tn = (2)n-1 What is the value of n? Explain. t9 = (2)9-1 t9 = 256 There would be 256 mg of venom after eight hours. 1.3.7

Assignment Suggested Questions Page 39: 1d,e,f, 2a, 3c, 5a,b, 6a, 7, 8, 10, 14 1.3.8