1. Algebra II Unit 1 Lesson 2, 3 & 5
Lesson 2: F Find the next term in a sequence described recursively.
Lesson 3: F Find a recursive expression for the general term in a sequence described recursively.
Lesson 5: Exhibit knowledge of geometric sequences.
Lesson 4: Solve complex arithmetic problems involving percent of increase /decrease or requiring integration of several concepts
(i.e. using several ratios, comparing percentages or averages).

2. Then define a sequence built by adding and subtracting as arithmetic and a sequence built by multiplying or dividing as geometric.

3. Do Now: Identify each sequence as arithmetic, geometric or neither.
14, 7, 3.5, 1.75,…
47, 41, 35, 29,…
1, 1, 2, 3, 5, 8,…
The non-horizontal cards
Did anyone find the arithmetic series in the card house?
Divide by 2
Subtract 6 (arithmetic)
Fibonacci
Card house: 8, 6, 4, 2 (ignoring the cards laid parallel to the floor because this doesn’t have to be a consistent amount) arithmetic – each layer gets two additional cards

4. According to the US EPA, each American produced an average of 978 lb of trash in 1960.
This increased to 1336 lb in By 2000, trash production had risen to 1646 lb/yr per person. How would you find the total amount of trash a person produced in a lifetime?
You would have to add the amount of trash produced per person each year the person was alive. Adding the numbers is finding the sum of a sequence – also known as a series.
EPA: Environmental Protection Agency. 𝑚_1 = 17.9 lbs./yr 𝑚_2 = 15.5 lbs./yr

5. Arithmetic Series Key Concepts:
We are learning about mathematical series and the summation notation used to represent them.
Write recursive and explicit formulas for the terms of the series and find partial sums of the series.
Series is an infinitely long expression, a summation, an indicated sum.
A mathematical series is different from a sequence because the terms are being added. It is not a sum because a sum is a number.
A series is convergent if the partial sums have a long-run value. This value is called the sum of the series, and the series converges to this number. Sometimes people call the number to which the partial sums converge “the infinite sum”

6. Vocabulary
Is the common meaning of the word series the same thing as the mathematical meaning?
In common usage, series means a sequence of events rather than a summation.
The word series is both plural and singular.
Partial sums: a sum of a finite number of terms of a series.
Finite: limited

7. Summation Symbol
When we don’t want to write out a whole bunch of numbers in the series, the summation symbol is used when writing a series. The limits are the greatest and least values of n.
Upper Limit (greatest value of n)
Explicit function for the sequence
Summation symbol
Lower Limit (least value of n)
So, the way this works is plug in n=1 to the equation and continue through n=3.
(5*1 + 1) + (5*2 +1) + (5*3 + 1) = 33

8. Understanding Sigma Notation
Evaluate the following expressions
Answers; 30, 65,-12

9. What did Gauss do?
The example of page 632 summarizes Gauss’s solution process

11. Examples Ex. 1.) Find the sum of the first 50 multiples of 6:
Ex.2.) Find the sum of the first 75 even numbers starting with 2.
u sub 50 = 300
The 75th even number is 150

12. Find the indicated valuesif
Highlight the similarities and differences in the questions.
The first question is asking for a sum of all the values from when k is 4 to when k is 18.
The second question wants the sum for the first nine values and is found using a similar process.
The third question is really only asking for one value; what is the value of u sub n when n is 18.
Answers
975
279 58

13. How many seats are there in all 35 rows of the concert hall?
The concert hall in the picture below has 59 seats in Row 1, 63 seats in Row 2, 67 seats in Row 3, and so on.
How many seats are there in all 35 rows of the concert hall?
Suppose that Seat 1 is at the left end of Row 1 and that Seat 60 is at the left end of Row 2. Describe the location of Seat 970.
There are 195 seats in the 35th row.
Use the partial sum formula to find a total of 4445 seats
Make a table to answer the last question
Row 12, third seat from the right

14. Practice
Page: 633 #(1 – 5)

15. Sum of a Finite Arithmetic Series
Let’s try one: evaluate the series:
5, 9, 13,17,21,25,29

16. Writing the given series in summation form.
Evaluate:
Yes, you can add manually. But let’s try using the shortcut:
n = 6
Rule: 2n + 100

17. Practice
Find the number of terms, the first term and the last term. Then evaluate the series:
Ex.1
Ex.2

18. Why is the answer not 58?
Note: this is NOT an arithmetic series. You can NOT use the shortcut; you have to manually calculate all values.
Notice we can use the shortcut here:

19. Arithmetic Sequence:
The difference between consecutive terms is constant (or the same).
The constant difference is also known as the common difference (d).
(It’s also the number you are adding every time!)

20. The general form of an ARITHMETIC sequence.
First Term:
Second Term:
Third Term:
Fourth Term:
Fifth Term:
a = u
As well.
nth Term:

21. Formula for the nth term of an ARITHMETIC sequence.
If we know any three of these we ought to be able to find the fourth.

22. Example: Decide whether each sequence is arithmetic.
5,11,17,23,29,…
11-5=6
17-11=6
23-17=6
29-23=6
Arithmetic (common difference is 6)
-10,-6,-2,0,2,6,10,…
-6-(-10)=4
-2-(-6)=4
0-(-2)=2
2-0=2
6-2=4
10-6=4
Not arithmetic (because the differences are not the same)

23. Rule for an Arithmetic Sequence
n = number of terms
an = last term
an= a1+(n-1)d

24. Example: Write a rule for the nth term of the sequence 32,47,62,77,…
Example: Write a rule for the nth term of the sequence 32,47,62,77,… . Then, find u12.
There is a common difference where d=15, therefore the sequence is arithmetic.
Use un=u1+u(n-1)d
un=32+(n-1)(15)
un=32+15n-15
un=17+15n
u12=17+15(12)=197

25. Example: One term of an arithmetic sequence is a8=50
Example: One term of an arithmetic sequence is a8=50. The common difference is Write a rule for the nth term.
Use an=a1+(n-1)d to find the 1st term!
a8=a1+(8-1)(.25)
50=a1+(7)(.25)
50=a1+1.75
48.25=a1
* Now, use an=a1+(n-1)d to find the rule.
an=48.25+(n-1)(.25)
an= n-.25
an=48+.25n

26. an=a1+(n-1)d an=-6+(n-1)(4) OR an=-10+4n
Example: Two terms of an arithmetic sequence are a5=10 and a30=110. Write a rule for the nth term.
Begin by writing 2 equations; one for each term given.
a5=a1+(5-1)d OR 10=a1+4d
And
a30=a1+(30-1)d OR 110=a1+29d
Now use the 2 equations to solve for a1 & d.
10=a1+4d
110=a1+29d (subtract the equations to cancel a1)
-100= -25d
So, d=4 and a1=-6 (now find the rule)
an=a1+(n-1)d
an=-6+(n-1)(4) OR an=-10+4n

27. Arithmetic Series Last Term
The sum of the terms in an arithmetic sequence
The formula to find the sum of a finite arithmetic series is:
1st Term
# of terms

28. Example: Consider the arithmetic series 20+18+16+14+… .
Find n such that Sn=-760

29. Always choose the positive solution!
-1520=n( n)
-1520=-2n2+42n
2n2-42n-1520=0
n2-21n-760=0
(n-40)(n+19)=0
n=40 or n=-19
Always choose the positive solution!

30. An introduction…………
Sequence Sum Sequence Sum
Arithmetic Sequences
Geometric Sequences
ADD
To get next term
MULTIPLY
To get next term

31. Vocabulary of Sequences (Universal)

32. Example: The nth Partial Sum
The sum of the first n terms of an infinite sequence is called the nth partial sum.
Example: The nth Partial Sum

33. Vocabulary of Sequences (Universal)

34. 1, 4, 7, 10, 13, ….
Infinite Arithmetic
No Sum
3, 7, 11, …, 51
Finite Arithmetic
1, 2, 4, …, 64
Finite Geometric
1, 2, 4, 8, …
Infinite Geometric
r > 1
r < -1
No Sum
Infinite Geometric
-1 < r < 1

35. Find the sum, if possible:

36. Find the sum, if possible:

37. Find the sum, if possible:

38. Find the sum, if possible:

39. Find the sum, if possible:

40. The Bouncing Ball Problem – Version A
A ball is dropped from a height of 50 feet. It rebounds 4/5 of
it’s height, and continues this pattern until it stops. How far
does the ball travel? 50 40 40 32 32
32/5
32/5

41. The Bouncing Ball Problem – Version B
A ball is thrown 100 feet into the air. It rebounds 3/4 of
it’s height, and continues this pattern until it stops. How far
does the ball travel?
100
100 75 75
225/4
225/4

42. Sigma Notation

43. UPPER BOUND
(NUMBER)
SIGMA
(SUM OF TERMS)
NTH TERM
(SEQUENCE)
LOWER BOUND
(NUMBER)

47. Rewrite using sigma notation: 3 + 6 + 9 + 12
Arithmetic, d= 3

48. Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1
Geometric, r = ½

49. Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4
Not Arithmetic, Not Geometric

50. Rewrite the following using sigma notation:
Numerator is geometric, r = 3
Denominator is arithmetic d= 5
NUMERATOR:
DENOMINATOR:
SIGMA NOTATION: