1. 1 Geometric Sequences MCR3U

2. 2 (A) Review A sequence is an ordered set of numbers. A sequence is an ordered set of numbers. An arithmetic sequence has a pattern to it => the constant difference between successive terms. An arithmetic sequence has a pattern to it => the constant difference between successive terms. Today, we will explore other sequences that have another pattern to them. Today, we will explore other sequences that have another pattern to them.

3. 3 (B) Geometric Sequences Comment upon any pattern you see in the sequences - i.e. what makes these sequences easy to work with?? Comment upon any pattern you see in the sequences - i.e. what makes these sequences easy to work with?? ex 1. 2,10,50,250,..... ex 1. 2,10,50,250,..... ex 2. 5,-10,20,-40,80,..... ex 2. 5,-10,20,-40,80,..... ex 3. 6, 0.6, 0.06, 0.006, 0.0006,.... ex 3. 6, 0.6, 0.06, 0.006, 0.0006,.... ex 4. 2,4,8,16,32,64,…. ex 4. 2,4,8,16,32,64,…. ex 5. 100, 50, 25, 12.5, 6.25, … ex 5. 100, 50, 25, 12.5, 6.25, … Each pair of successive terms have a constant ratio => thus making them Geometric Sequences Each pair of successive terms have a constant ratio => thus making them Geometric Sequences

4. 4 (C) Representing the Geometric Sequences (1) Table of Values (1) Table of Values from which we notice no common first difference from which we notice no common first difference but if we divide each term by the preceding term, we notice a common ratio but if we divide each term by the preceding term, we notice a common ratio Time012345 Amount248163264

5. 5 (C) Representing the Geometric Sequences (2) Scatter plots (2) Scatter plots from which we notice a curved relation from which we notice a curved relation

6. 6 (C) Representing the Geometric Sequences (3) Equations and Formulas (3) Equations and Formulas But how do we determine the formula that generates the terms of the sequence?? But how do we determine the formula that generates the terms of the sequence??

7. 7 (D) The General Term of a Geometric Sequences Consider the following analysis: Consider the following analysis: t 1 = 3 = 3 x 1 = 3 x 2 0 t 1 = 3 = 3 x 1 = 3 x 2 0 t 2 = 6 = 3 x 2 = 3 x 2 1 t 2 = 6 = 3 x 2 = 3 x 2 1 t 3 = 12 = 3 x 4 = 3 x 2 x 2 = 3 x 2 2 t 3 = 12 = 3 x 4 = 3 x 2 x 2 = 3 x 2 2 t 4 = 24 = 3 x 8 = 3 x 2 x 2 x 2 = 3 x 2 3 t 4 = 24 = 3 x 8 = 3 x 2 x 2 x 2 = 3 x 2 3 t 5 = 48 = 3 x 16 = 3 x 2 x 2 x 2 x 2 = 3 x 2 4 t 5 = 48 = 3 x 16 = 3 x 2 x 2 x 2 x 2 = 3 x 2 4 We can see a pattern emerging as to how to calculate the general term of a geometric sequence as: t n = ar n-1, where a is the first term of the sequence, n is the term number, and r is the common ratio. We can see a pattern emerging as to how to calculate the general term of a geometric sequence as: t n = ar n-1, where a is the first term of the sequence, n is the term number, and r is the common ratio.

8. 8 (D) The General Term of an Arithmetic Sequences Working with the formula t n = ar n-1, we notice two things: Working with the formula t n = ar n-1, we notice two things: If r > 1, then the terms increase If r > 1, then the terms increase If 0 < r < 1, then the terms decrease If 0 < r < 1, then the terms decrease

9. 9 (E) Examples ex 1. Write the first 6 terms of the sequence defined by t n = 5(-2) n-1 ex 1. Write the first 6 terms of the sequence defined by t n = 5(-2) n-1 ex 2. Given the formula for the nth term as t n = -5(4) n-1, find 10 th term. ex 2. Given the formula for the nth term as t n = -5(4) n-1, find 10 th term. ex 3. Find the formula for the nth term given the geometric sequence 2,6,18,...... Then find the 7 th term. ex 3. Find the formula for the nth term given the geometric sequence 2,6,18,...... Then find the 7 th term. ex 4. How many terms are there in the geometric sequence 3,6,12,....,384 ex 4. How many terms are there in the geometric sequence 3,6,12,....,384 ex 5. If the 5 th term of a sequence is 1875 and the 7 th term is 46,875, find a, r, and t n and the first three terms of the sequence. ex 5. If the 5 th term of a sequence is 1875 and the 7 th term is 46,875, find a, r, and t n and the first three terms of the sequence.

10. 10 Examples ex 6. Since 1967, the average annual baseball salary was $19,000. The average annual salary has been rising at a rate of 17% per year. Determine the equation for geometric sequence and then predict the average annual salary for 2007. ex 6. Since 1967, the average annual baseball salary was $19,000. The average annual salary has been rising at a rate of 17% per year. Determine the equation for geometric sequence and then predict the average annual salary for 2007.

11. 11 (F) Internet Links College Algebra from WTAMU College Algebra from WTAMU (http://www.wtamu.edu/academic/anns/mps/m ath/mathlab/col_algebra/col_alg_tut54c_arit h.htm)

12. 12 Homework Complete text pg 392-393 #1-5, 7-9. Complete text pg 392-393 #1-5, 7-9.