Arithmetic Sequences and Series Section 9-2

2 Objectives Use sequence notation to find terms of any sequence Use summation notation to write sums Use factorial notation Find sums of finite and infinite series

3 9.1 Sequences & Series SEQUENCE: A list that is ordered so that it has a 1 st term, a 2 nd term, a 3 rd term and so on. example: 1, 5, 9, 13, 17, … a 1 = 1; a 2 = 5; a 3 = 9, etc. The nth term is denoted by: a n The nth term is used to GENERALIZE about other terms.

4 The three dots mean that this sequence is INFINITE. example: 1, 5, 9, 13, 17, … example: 2, -9, 28, -65, 126 This is a FINITE sequence.

5 Given a “rule” for a sequence, find the sequence terms. EXAMPLE 1:

6 Write the first four terms of the sequence whose nth term is given by:

7 Recursion Formula Defines the nth term of a sequence as a function of the previous term.

8 Find the first four terms

9 Factorial Notation n! = n(n – 1)(n – 2)…1 Special case: 0! = 1 8 Math/prb/4/enter = 40,320

10 A sequence is arithmetic if the differences between consecutive terms are the same. 4, 9, 14, 19, 24,... 9 – 4 = 5 14 – 9 = 5 19 – 14 = 5 24 – 19 = 5 arithmetic sequence The common difference, d, is 5.

11 Example: Find the first five terms of the sequence and determine if it is arithmetic. a n = 1 + (n – 1)4 This is an arithmetic sequence. d = 4 a 1 = 1 + (1 – 1)4 = = 1 a 2 = 1 + (2 – 1)4 = = 5 a 3 = 1 + (3 – 1)4 = = 9 a 4 = 1 + (4 – 1)4 = = 13 a 5 = 1 + (5 – 1)4 = = 17

12 The nth term of an arithmetic sequence has the form a n = dn + c where d is the common difference and c = a 1 – d. The nth Term of an Arithmetic Sequence 2, 8, 14, 20, 26,.... d = 8 – 2 = 6 a 1 = 2 c = 2 – 6 = – 4 The nth term is: a n = 6n – 4.

13 a 1 – d = Example: Formula for the nth Term Example: Find the formula for the nth term of an arithmetic sequence whose common difference is 4 and whose first term is 15. Find the first five terms of the sequence. a n = dn + c = 4n , d = 4 a 1 = 15 19, 23, 27, 31. The first five terms are 15 – 4 = 11

14 The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with n terms is given by = ? n = 10 a 1 = 5a 10 = 50

15 Example: The nth Partial Sum The sum of the first n terms of an infinite sequence is called the nth partial sum. a 1 = – 6 a n = dn + c = 4n – 10 Example: Find the 50th partial sum of the arithmetic sequence – 6, – 2, 2, 6,... d = 4c = a 1 – d = – 10 a 50 = 4(50) – 10 = 190

16 The sum of the first n terms of a sequence is represented by the summation notation Where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation. Summation Notation

17 Definition of Series Consider the infinite sequence a 1, a 2, a 3,..., a i, The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence. a 1 + a 2 + a a n 2.The sum of all the terms of the infinite sequence is called an infinite series. a 1 + a 2 + a a i +...

18 Example Expand and evaluate the sum: Solution:

19 Example: Summation Notation Example: Find the partial sum. a1a1 a 100

20 Homework WS 13-4