Warm Up ChallengeLet’s see who will be the first to get the correct answer.

Without a calculator –

Add up all the integers between 1 and 10.

1+2+3+4+……+10 =

…Now do the same thing up to 100.

A StoryIn the late 1700’s a school teacher in a poor German town wanted to find a way to keep the kids busy for a half hour. He gave them the task of adding all the numbers from 1 to 100 and reporting back with the answer.

Less then 5 minutes after the students began working, a 10 year old boy came up to the teacher with the correct answer of 5050.

How did he do it?

Let’s look at the numbers between 1 and 10 again and see if we can find a pattern or a trick.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Let’s try to look at the distance between the numbers:• 1 +10 = 11• 2+9 = 11• 3+8 = 11• 4+7 = 11• 5+6 = 11

• We have added up the possible sets of numbers and got 5 sets that add to 11 or : 11(5) = 55

So the sum of the first 10 numbers is 55.

For a 100 numbers we could look at a similar situation:

1 + 100 = 1012 + 99 = 1013 + 98 = 1014 + 97 = 1015 + 96 = 101...100 + 1 = 101

We can line up all the numbers forwards….…and backwards, 100 to 1….…..then adding the numbers up always produced the same number, 101.

Multiplying 101 by the 100 numbers he was supposed to add up gave him the answer of 10100.

But since we added each number ( 1 to 100) twice, dividing 10100 by two gives the correct answer of 5050

Carl Friedrich Gauss

What is a Series?A series is the___________________________________

For instance; the sequence of odd numbers: 1,3,5,9,11,….

Can be expressed as a series: 1+3+5+9+11…… - Try to take the sum of the first few terms; what

interesting observation can you make?

We will be talking about two kinds of series:

1) ______________________________________

2) ______________________________________

sum of numbers in a given sequence

Arithmetic Series – Day 1/Day 2

Geometric Series – Day 2/Day3

Easy ExampleWhat is the sum of the first 6 terms of a geometric sequence.

• Write the sequence described as a series: 2+4+6+8+10+12

• Putting this in a calculator gives us our answer of 42

However, for longer sequences, it is helpful to convey information about a series in concise way without having to have a long string of addition.

Summation NotationIf a series has a lot of terms it is inconvenient to write all the terms out.

For example, if the following series has 10 terms, you could write it out as :1+3+5+7+9+11+13+15+17+19

However, there is a more convenient way to do this:

_________________ __________________________

____________________

Lower limit

Upper limit

Explicit Formula for sequence

In English, this means “add up all the terms in the formula starting from n equals one and continue until n equals 10. “ The symbol is the capitol Greek letter ‘Sigma’.

∑𝑛=1

5

𝑛 1

4

3

2

5

+

+

+

+

Start an n=1

Stop when n=5

Closer Look atSummation Notation

ExamplesWrite the following sequence in summation notation:

1) 1+2+3+4+……..n=100 2) 5+10+15+20+……..n=10 _____________

3) 3+7+11+15+…..n=15

Common DifferenceIn the terms

Remember linear functions: y=mx+b?The b term comes from adjusting the common difference in the sequence to make it represent the first term correctly.

Examples1) 3+4+5+6+……..n=20

2) 5+ 2+ (-1) + (-4)…….n=8

3) 1+4+7+10 +….n=11

Notice how all the numbers are 2 more then multiples of one.

On a Quarter Sheet of paper1) Choose a number between -20 and 20: ____________________

2) Choose a number between – 50 and 50: ___________________

3) Choose a number between 10 and 20: _____________________

Write an arithmetic Series of the first 4 terms where:• The first term is the number you wrote in #1• The common difference is the number you wrote in #2• The Number of terms is the number you wrote in #3

Crumple up your paper and…..

Snowball Fight!

Find a paper on the floor and:

1) Check that their arithmetic series is correct and actually is an arithmetic sequence.

2) Look at the number of terms the person has given (in #3) and write the sequence in summation notation.

Using Summation NotationYou can use summation notation to find out a number of things about a series including:

1) _____________________

2) _____________________

3) _____________________

4) _____________________

First term

Last term

Number of terms

Sum of a series

Using Summation NotationFor the series given, find the number of terms, the first term, and the last term. Then evaluate the series.

1) The number of terms can always be expressed as:

___________________________________________

2) The first term can be attained by plugging the lower limit into the formula:

first term:________________________

3) The last term can be obtained by plugging the upper bound into the formula:

Last term:_______________________________

(Upper Limit – Lower Limit) +1 → (6-1)+1= 6

3(1) + 3 = 6

𝑎𝑛=3 (6 )+3=18+3=21

Using Summation Notation4) The formula for the first n terms of an arithmetic sequence, starting with n = 1, is:

(+

Hence, since we have:

Where _____ and = _______ and n = ______Then to evaluate the series:

_______________________________________________

You can check your answer as well (since there are only 6 terms, it’s relatively easy)(6+21

6 21 6

6+9+12+15+18+21 = 81

Read this as: “the sum of the first n terms”

∑𝑛=1

23

4𝑛−2

Find the sum of the following series:

Another Example

Recall: The sum of the first n terms of an odd number series is always a square number.

For example: 1+3+5+….+19 =

→

and

My question: If I didn’t start at 1 as my lower bound will I still have a square number.

For example if I started a series on the 5’th odd number and ended on the 20’th odd number will the sum still be an odd number?

∑1

10

2𝑛−1

Take the sum of the first 100 integers but skip all the numbers that are multiples of 3’s and 5’s in the series.

For example:_______________________________

https://www.youtube.com/watch?v=PHxvMLoKRWg

Mr. D. drops a penny from an airplane at 16,000 feet. The first second the penny falls 6 feet, the second second it falls 38 feet, the third second it falls 70 feet During second four it falls 102 feet. Will the penny hit the ground in 30 seconds? (To keep it simple we will assume the penny falls the same rate the whole time and terminal velocity is not reached)

Challenge question: How many seconds will it take for the penny to hit the ground. Calculate this to the nearest tenth of a second without using the brute force method.

A basic law of physics is that as objects fall they pick up speed until they hit a terminal velocity.