1. A LESSON BY U S PRAJAPATI, PGT MATH, KV KHAGAUL GEOMETRIC SEQUENCES AND SERIES

2. 2 DEFINITION OF SEQUENCE An infinite sequence is a function whose domain is the set of positive integers. a 1, a 2, a 3, a 4,..., a n,... The first three terms of the sequence a n = 2n 2 are a 1 = 2(1) 2 = 2 a 2 = 2(2) 2 = 8 a 3 = 2(3) 2 = 18. finite sequence terms

3. 3 DEFINITION OF GEOMETRIC SEQUENCE A sequence is geometric if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512,... geometric sequence The common ratio, r, is 4.

4. 4 THE NTH TERM OF A GEOMETRIC SEQUENCE The nth term of a geometric sequence has the form a n = a 1 r n - 1 where r is the common ratio of consecutive terms of the sequence. 15, 75, 375, 1875,... a 1 = 15 The nth term is 15(5 n-1 ). a 2 = 15(5) a 3 = 15(5 2 ) a 4 = 15(5 3 )

5. 5 EXAMPLE: FINDING THE NTH TERM Example: Find the 9th term of the geometric sequence 7, 21, 63,... a 1 = 7 The 9th term is 45,927. a n = a 1 r n – 1 = 7(3) n – 1 a 9 = 7(3) 9 – 1 = 7(3) 8 = 7(6561) = 45,927

6. COPYRIGHT © BY HOUGHTON MIFFLIN COMPANY, INC. ALL RIGHTS RESERVED. 6 DEFINITION OF SUMMATION NOTATION The sum of the first n terms of a sequence is represented by summation notation. index of summation upper limit of summation lower limit of summation

7. 7 THE SUM OF A FINITE GEOMETRIC SEQUENCE The sum of a finite geometric sequence is given by 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a 1 = 5

8. 8 DEFINITION OF GEOMETRIC SERIES The sum of the terms of an infinite geometric sequence is called a geometric series. a 1 + a 1 r + a 1 r 2 + a 1 r 3 +... + a 1 r n-1 +... If |r| < 1, then the infinite geometric series has the sum

9. 9 EXAMPLE: SUM OF INFINITE GEOMETRIC SERIES Example: Find the sum of The sum of the series is