Advanced Algebra notes about the arithmetic series

1. Let’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.

2. In 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?

3. 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.

4. For a 100 numbers we could look at a similar situation:
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
4 + 97 = 101
5 + 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

5. 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

6. What 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.

7. If 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:
𝑛=1
10
2𝑛 − 1
_________________ __________________________
____________________
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’.

8. 𝑛=1
5
𝑛 1
4
3
2
5
+
+
+
+
Start an n=1
Stop when n=5

9. Write the following sequence in summation notation:
1) 1+2+3+4+……..n=100
𝑛=1
100
1𝑛
2) 5+10+15+20+……..n=10 _____________
𝑛=1
10
5𝑛
3) 3+7+11+15+…..n=15
𝑛=1
15
4𝑛 − 1
Common
Difference
In 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.

10. 1) 3+4+5+6+……..n=20
𝑛=1
20
𝑛 + 2
2) 5+ 2+ (-1) + (-4)…….n=8
𝑛=1
8
𝑛 + 7
3) 1+4+7+10 +….n=11
𝑛=1
11
3𝑛 − 2
Notice how all the numbers are 2
more then multiples of one.

11. 1) 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…..
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.

12. You 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

13. For the series given, find the number of terms, the first term, and the last term. Then evaluate the series.
1
6
3𝑛 + 3
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
𝑎1 = 3(1) + 3 = 6
𝑎 𝑛 = 3 6 + 3 = 18 + 3 = 21

14. 4) The formula for the first n terms of an arithmetic sequence, starting with n = 1, is:
𝑆 𝑛 =
𝑛
2
(𝑎1+𝑎 𝑛)
Hence, since we have:
1
6
3𝑛 + 3
Where 𝑎1 = _____ and 𝑎 𝑛= _______ and n = ______
Then to evaluate the series:
_______________________________________________
You can check your answer as well (since there are only 6 terms, it’s relatively easy)
𝑆10 =
6
2
(6+21) = 81
6 21 6
6+9+12+15+18+21 = 81
Read this as: “the sum of the first n terms”

15. 𝑛=1
23
4𝑛 − 2
Find the sum of the following series:

16. Recall: The sum of the first n terms of an odd number series is always a square number.
For example:
1+3+5+….+19 =
→
10
2
1 + 19 = 100
and 100 = 10
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

17. 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:_______________________________

18. 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.