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

PracticePage 140, # 1, 3, 5

Look at the graph

of the sequence:

2, 4, 6, 8, 10

Arithmetic Sequence

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

11

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

Arithmetic Sequence

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 = 1/2 - 1/4n; 2

an = 6n – 10; 50

Problem 11 – 15 is like finding the linear equation given two points an = a1 + d(n – 1)1. Find the common difference – d (slope)

2. Substitute a point and solve for a1

3. Plug common difference and a1 into the general equation and simplify

# 11

# 13

# 15

an = 14n – 40

an = -5 - n

an = n/4 + 2

HomeworkPage 140, # 2 – 16 even

Finding the Sum of an Arithmetic SequenceThe 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):

2

1 nn

aanS

Practice – page 140# 17

# 19

# 21

# 23

100

-210

an = n - 2

an = -n/2 + 5

2

1 nn

aanS

HomeworkPage 140, # 2 – 24 even