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.
Arithmetic Sequence • An arithmetic sequence is nothing more than alinear 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”
Practice • Page 140, # 1, 3, 5
Arithmetic Sequence • Look at the graph of the sequence: 2, 4, 6, 8, 10
Arithmetic Sequence • Let’s take the point-slope linear form (y – y1) = m(x –x1) Solving for y , and calling it f(x) gives: f(x) = m(x –x1) + y1 • The terms of a sequence are the outputs of some function , so f(x) = an an = m(x –x1) + y1
an = m(x –x1) + y1 • The domain of a sequence is usually the natural numbers. Let's use n for them. So, x = n in our formula. an = m(n –x1) + y1
an = m(n –x1) + y1 • 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. an = d(n –x1) + y1
an = d(n –x1) + y1 • The first term is always labeled a1. It is the ordered pair (1, a1). We'll use it for the (x1, y1) point in the point-slope form. • Putting them all together we have a rule for creating nth term formula: an = d(n – 1) + a1
Rule for nth term formula: an = a1 + d(n – 1) Where: an is value of the nth term d is the “common difference” n is the number of terms a1 is the first term NOTE: Be sure to simplify NOTE: Look at this on a graph
Practice – page 140 an = a1 + d(n – 1) • # 7 • # 9 an = 6n – 10; 50 an = 1/2 - 1/4n; 2
Problem 11 – 15 is like finding the linear equation given two points an = a1 + d(n – 1) • Find the common difference – d (slope) • Substitute a point and solve for a1 • Plug common difference and a1 into the general equation and simplify • # 11 • # 13 • # 15 an = 14n – 40 an = -5 - n an = 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 • # 23 100 -210 an = n - 2 an = -n/2 + 5
Homework • Page 140, # 2 – 24 even