Download INT Unit 3 Notes

Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 3 Notes
1-7 Homework Notes
1-7: Explicit Formulas for Sequences
A _____________________ of numbers always have a pattern that can be explained by a formula.
Example: Draw out the sequence of dots on p. 42 below.
The number of dots in each of the figures above produces a _______________________ of numbers. Write
these numbers below.
Each number in the sequence is called a ________. So the first term is ______, the second term is _____, etc.
Example: What would be the sixth term in the triangular number sequence?
If we think of the triangles as half of a rectangle (see pictures on p. 43), the height of the rectangle would be n
(the number of the term) and the base would be n + 1. Therefore, we can think of using the formula
________________________________
Where t(n) = the nth triangular number. The sentence above is called an ____________________________________ for
the nth term of the sequence 1,3,6,10,15,21… because we use it to calculate the _________ term __________________
by substituting a value for n.
Example: What is the 20th triangular term?
A special notation is often used with sequences. Instead of using t(10) = 55 to indicate that the 10th term is 55, we write
______________ which is read “______________________________________”. The number 10 is called a
___________. This is often called an _________ because it indicates the position of the term in its sequence.
Complete p. 45 #5-9
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 3 Notes
1-7 In-Class Notes
1-7: Explicit Formulas for Sequences
Example: Consider the formula tn = 3n, for integers n≥1. What are the first four terms of the sequence it
defines? Evaluate t20 and explain in words what it represents.
Example: A microbe reproduces by splitting to make 2 cells. Each of these cells then splits in half to make a total of 4
cells. Each of these splits to make a total of 8, and so on. Each splitting is called a generation. If a colony begins with
500 microbes, the following equation gives the number of microbes in the nth generation (assuming no microbes die).
Pn = 500(2)n-1
Complete p. 46 #13-15,17-21 on a separate sheet.
1-8 & 1-9 Homework Notes
1-8: Recursive Formulas for Sequences
If we have the formula an = -3 + 5n, write the first 5 terms of this sequence.
We can quickly generate this sequence using a calculator. Complete the following keystrokes on your
calculator and copy the answers it gives you below.
Keystroke
Answer
Keystroke
2 (enter)
+ 5 (enter)
+ 5 (enter)
+ 5 (enter)
+ 5 (enter)
+ 5 (enter)
Answer
Name:
Hour:
Why did we decide to add 5 each time?
Mrs. Gorsline
Integrated Math 2
Unit 3 Notes
Your calculator uses “ANS” when you want it to use the answer from the previous line. In “sequence speak” we call this
the previous term.
A ______________________________ for a sequence is a set of statements that
a. indicates the _____________________ or _______________________ and
b. tells how the ________ term is related to one or more of the _____________________________________
So the sequence 40,20,10,5,2.5 … could be described recursively as
However, remember that we use _____ to describe the first term and we use _______ to describe the nth term.
Complete p. 53 #1,2,6,7
1-9: Notation for Recursive Formulas
Instead of writing “previous term” or “ANS”, we use a different notation since we always want to use variables in
algebra. If tn is the term that we want, then the previous term should be denoted as ____________.
Example: Consider the following sequence. Describe the sequence in words, then find the first 5 terms.
t1 = 60
tn = 2 ∙ tn-1, for integers n≥1
Complete p. 58 #1,4,5
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 3 Notes
1-8 & 1-9 In-Class Notes
Consider the sequence defined recursively as
T1 = 1
Tn = Tn-1 + n, for integers n≥2
Find T2, T3, and T4.
Example: A male bee develops from an unfertilized egg. That is, a male bee has a mother, but no father. A female bee
develops from a fertilized egg. That is, a female bee has both a mother and a father. See the figure on p. 57 that shows
the ancestors of a male bee. Counting the symbols in each row gives the number of bees in each generation. The
number of bees in the first six generations is 1,1,2,3,5,8…
Use the first two terms to write a recursive sequence of this situation. Then find the 7th, 8th, and 9th terms.
This sequence has a special name. It is called the __________________________________________, and it is found
throughout nature. You will be researching this man at the end of this unit.
Complete p. 53, #11,12,15,16 and p. 58, # 1,2,4,7,9,11,13,14,17 on a separate sheet.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 3 Notes
3-7 & 3-8 Homework Notes
3.7: Recursive Formulas for Arithmetic Sequences
A sequence with a constant difference is called an __________________________________________. A
constant difference means that you are _______________ or ________________________ the same number
over and over again.
Example: Write the recursive formula for the sequence below:
1000, 4000, 7000, 10000, 13000, 16000 …
This shows that the difference between the _______ term and the ___________ term is a ________________.
Theorem: The sequence defined by the recursive formula
Is the ___________________________ with first term ____ and constant difference ___.
Example: Consider the sequence generated by
a1 = 2000
an = an-1 + 40, for integers n≥2
Describe this sequence in words. Then write the first five terms.
Read p. 177 carefully (program). We will do this together in class.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 3 Notes
3.8: Explicit Formulas for Arithmetic Sequences
Let’s look at that sequence again: 1000, 4000, 7000, 10000, 13000, 16000 … with recursive formula
a1 = 1000
an = an-1 + 3000, for integers n≥2
Suppose you wanted the 50th term of this sequence. Obviously this would be difficult to find with only the
recursive sequence above. We need to find the _________________ formula.
Example: Fill in the following table. Use that information to find the explicit formula and then find a50 and
explain what it means in words.
Number of
term
1
2
3
4
5
6
7
8
n
term
Theorem: The _______ term of an arithmetic sequence with first term _______ and constant difference ___ is
given by the explicit formula __________________________________.
Example: Find the 40th term of the arithmetic sequence 100, 97, 94, 91 ….
Read the bottom of p. 182 carefully (program). We will do this together in class.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 3 Notes
3-7 & 3-8 In-Class Notes
Example: Briana borrowed $370 from her parents for airfare to Europe. She will pay them back at the rate of
$30 per month. Let an be the amount she still owes after n months. Find a recursive formula for this
sequence.
a1 =
an =
Graphs of arithmetic sequences should always be _________________________ because there is either a constant
____________________ or a constant _______________ situation.
Example: In a concert hall the first row has 20 seats in it, and each subsequent row has 2 more seats than the
row in front of it. If the last row has 64 seats, how many rows are in the concert hall?
Programs for the TI-Nspire (attached worksheets) and the Ti-84 (below)
Recursive sequence:
Explicit Sequence:
: 1000A
_________________________
: For(N,1,6)
_____________________________
: For(N,1,6)
_________________________
: 3000N – 2000 A
_____________________________
: Disp A
_________________________
: Disp A
_____________________________
: End
_____________________________
: A + 3000A _________________________
: End
_________________________
Complete on a separate sheet: p. 177 #3,6-8,11,14 and p. 183 #4-6,11
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 3 Notes
7-5 Homework Notes
In an ______________________ sequence, each term after the first is found by ______________ a constant
difference to the previous term. If, instead, each tern after the first is found by ____________________ the
previous term by a constant, then a ___________________________ sequence is formed. For example:
48, 72, 108, 162, 243, 364.5...
Has the first term ____________ and a _______________________________ of 1.5.
The sequence defined by the recursive formula
Where ____ is a nonzero constant, is the
_____________________, or exponential sequence
With first term _____ and constant multiplier ____________. r is also known as the ____________________________.
Example: Give the first six terms and the constant multiplier in the geometric sequence
g1 = 3
gn = 5 gn-1
for int. n≥2
Explicit Formula for a geometric sequence
In the geometric sequence with first term ___________ and constant ratio r, ___________________________
for integers n ≥ 1.
Notice that in the explicit formula, the exponent of the ______ term is _______. When you substitute 1 for n
to find the first term, the constant multiplier has an exponent of ____________.
Complete p. 447 #2,7
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 3 Notes
7-5 In-Class Notes
Example: Write the first five terms of the sequence defined by gn = 8(-5)n-1
Example: Suppose a ball is dropped from a height of 5 meters, and it bounces up to 90% of its previous height after each
bounce. Let hn be the maximum height of the ball after the nth bounce. Find an explicit formula and find the maximum
height of the ball after the 10th bounce.
Geometric Sequences got their name because of special geometric patterns called ________________. When
you take the lengths of the sides of the subsequent shapes, you end up with a sequence.
So, as an overview of sequence formulas:
HW: Complete p. 447 #10,11,12,14 on a separate sheet