Electronic Devices If I see an electronic device other than a calculator (including a phone being used as a calculator) I will pick it up and your parents can come an get it.

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.

 An arithmetic sequence is nothing more than a linear function with the specific domain of the natural numbers. The outputs of the function create the terms of the sequence.  The difference between any two terms of an arithmetic sequence is a constant, and is called the “common difference” Arithmetic Sequence

Practice  Page 140, # 1, 3, 5

 Look at the graph of the sequence: 2, 4, 6, 8, 10 Arithmetic Sequence

 Let’s take the point-slope linear form (y – y 1 ) = m(x –x 1 ) Solving for y, and calling it f(x) gives: f(x) = m(x –x 1 ) + y 1  The terms of a sequence are the outputs of some function, so f(x) = a n a n = m(x –x 1 ) + y 1 Arithmetic Sequence

a n = m(x –x 1 ) + y 1  The domain of a sequence is usually the natural numbers. Let's use n for them. So, x = n in our formula. a n = m(n –x 1 ) + y 1

 The value m is the slope in a linear function. In the sequence world as we go from term to term, we find that the change in input is always 1 while the change in output never changes. It is common to all consecutive pairs of terms. In the sequence world the slope is exactly the same as the common difference, d. Then m = d. a n = d(n –x 1 ) + y 1

 The first term is always labeled a 1. It is the ordered pair (1, a 1 ). We'll use it for the (x 1, y 1 ) point in the point-slope form.  Putting them all together we have a rule for creating n th term formula: a n = d(n – 1) + a 1

Rule for n th term formula: a n = a 1 + d(n – 1) Where: a n is value of the n th term d is the “common difference” n is the number of terms a 1 is the first term NOTE: Be sure to simplify NOTE: Look at this on a graph

Practice – page 140 a n = a 1 + d(n – 1)  # 7  # 9 a n = 1/2 - 1/4n; 2 a n = 6n – 10; 50

Problem 11 – 15 is like finding the linear equation given two points a n = a 1 + d(n – 1) 1. Find the common difference – d (slope) 2. Substitute a point and solve for a 1 3. Plug common difference and a 1 into the general equation and simplify  # 11  # 13  # 15 a n = 14n – 40 a n = -5 - n a n = n/4 + 2

Homework  Page 140, # 2 – 16 even

Finding the Sum of an Arithmetic Sequence  The expression formed by adding the terms of an arithmetic sequence is called an arithmetic series.  The sum of the first n terms of an arithmetic series is: (Determine the equation via a spreadsheet):

Practice – page 140  # 17  # 19  # 21  # a n = n - 2 a n = -n/2 + 5

Homework  Page 140, # 2 – 24 even