Sequences and Series 4.7 & 8 Standard: MM2A3d Students will explore arithmetic sequences and various ways of computing their sums. Standard: MM2A3e Students will explore sequences of Partial sums of arithmetic series as examples of quadratic functions.

Where am I going to use this? Series and sequences are used in higher math to do approximations for problems that do not have exact answers.

Where am I going to use this? A more immediate use: A field house has a section where the seating can be arranged so the first row has 11 seats, the second row has 15 seats, the third row has 19 seats and so on. If there is sufficient space for 30 rows in the section, how many seats are in the last row of the section and in total? Last row has127 seats, and there are 2,070 seats total in the section – later.

Sequences and Series 4.7 A sequence is a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1. The values in the range are called the terms of the sequence. A finite sequence has a limited number of terms. An infinite sequence continues without stopping.

A sequence can be specified by an equation, or rule. Sequences and Series 4.7 A sequence can be specified by an equation, or rule.

Writing Terms of a Sequence Given the following rules, write the first six terms of each sequence: an = n – 2 ___________________________________________________________________________________________ an = 5n – 3 ____________________________________________________________________________________________ an = 2n __________________________________________________________________________________________________________________________________________ . -1, 0, 1, 2, 3, 4 2, 7, 12, 17, 22, 27 2, 4, 8, 16, 32, 64

Practice – page 135: # 1 # 3 # 5 7, 10,15, 22, 31, 42 9, 27, 81, 243, 729, 2187 -4/3, -2/3, -4/9, -1/3, -4/15, -2/9

Write Rules for Sequences We can observe data and generate a rule. Describe the pattern, write the next term, and write a rule for the nth term of the following sequences: 3, 6, 9, 12 . an = 1/(2n-1) . an = (n+1)/(n+2) an = 3n

Class Work: What is the rule for the opening example? (11, 15, 19, …) What is a30? a30 = 4*30 + 7 = 127, as noted before Page 135, # 11 an = 4n + 7 an = 2n + 1

Homework Page 135 # 2, 4, 6, 10, 12

Sequences and Series When the terms of a sequence are added together, the resulting expression is a series. Summation notation, or sigma notation, is used to write a series. For example, in the series i is the index of summation, 1 is the lower limit of summation, and 4 is the upper limit of summation.

Rule for a Linear Sequence: But the change in input = 1, so:

Writing Series Using Summation Notation Write the series using summation notation: 5 + 6 + 7 +…+ 12 4 + 8 + 12 + … 1/3 + 2/4 + 3/5 + … + 12/14 5/4 + 7/9 + 9/14 + 11/19 + 13/24

Class Work page 135: # 19 # 21 # 23 # 25 50 The lower limit is zero, so the first term should be 4; 4 + 7 + 10 + 13 + 16 = 50

Homework: Page 135, # 20 – 26 even