Grade 11 Functions (MCR3U) Unit 5: Pascal’s Triangle, Binomial Theorem, Sequences & Series Geometric Sequences Mr. Choi © 2018 E. Choi – MCR3U - All Rights Reserved

Geometric Sequence {a, ar, ar 2, ar 3,...} A sequence like 3, 9, 27, 81,…, where the ratio between consecutive terms is a constant, is called a geometric sequence. In a geometric sequence, the first term t1, is denoted as a. Each term after the first is found by multiplying a constant, called the common ratio, r, to the preceding term. The list then becomes . {a, ar, ar 2, ar 3,...} Geometric Sequences © 2018 E. Choi – MCR3U - All Rights Reserved

Geometric Sequences Formulas In general: {a, ar, ar 2, ar 3,...,ar n-1 ,...} Geometric Sequences © 2018 E. Choi – MCR3U - All Rights Reserved

Example 1 – Finding Formula for the nth term In the geometric sequences: {5, 15, 45,...}, find a) b) c) c) b) a) 10 n 10 n n n 5 5 Geometric Sequences © 2018 E. Choi – MCR3U - All Rights Reserved

Example 2 – Finding Formula for the nth term Given the geometric sequence: {3, 6, 12, 24, ...}. Find the term Which term is 384? We know the nth term is 384 !! b) a) n n 14 14 Drop the bases!! Geometric Sequences © 2018 E. Choi – MCR3U - All Rights Reserved

Example 3 – Find the terms in the sequence In a geometric sequence, t3 = 20 and t6 = -540. Find the first 6 terms of the sequence. (2) (1) Substitute into (1) (1) (2) Therefore the first 6 terms of the sequences are: Geometric Sequences © 2018 E. Choi – MCR3U - All Rights Reserved

Example 4 – Find the terms in the sequence In a geometric sequence, t3 = 20 and t6 = -540. Find the first 6 terms of the sequence. METHOD 2 To find a, we use the same thinking process!! t1 = 20r(1-3) tn=20r (n-3) Therefore the first 6 terms of the sequences are: Geometric Sequences © 2018 E. Choi – MCR3U - All Rights Reserved

Example 5 – Applications of Geometric sequence Determine the value of x such that Form a geometric sequence. Find the sequences and Therefore the sequences are: 5+4, 2(5)+5, 4(5)+5, ... 9, 15, 25,... Geometric Sequences © 2018 E. Choi – MCR3U - All Rights Reserved

Homework: Text Book: P. 392 #1,2bcefh,3-4,5bcdf,6-9,11,14-20 Work Sheet: Geometric Sequences Check the website for updates Geometric Sequences © 2018 E. Choi – MCR3U - All Rights Reserved

End of Lesson Geometric Sequences © 2018 E. Choi – MCR3U - All Rights Reserved