Sequences, Series, and Probability. Infinite Sequences and Summation Notation.
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Transcript of Sequences, Series, and Probability. Infinite Sequences and Summation Notation.
Infinite Sequence
An infinite sequence is a function whose domain is the set of positive integers.
Notation:
Hint: It’s the natural numbers from 1 to .
Examples:
Find the first 3 terms of the sequence, then find the 10th and 15th terms of the sequence.
1)
2)
Recursive Sequences
A recursive sequence is when you are given the first term of the sequence , and each term after is acquired by using the previous term.
Ex: Find the first 4 terms and the nth term for the following infinite sequence defined recursively:
Bell Work:
Find the first four terms and the 15th term for the sequence below:
1)
Find the first four terms and the nth term for the recursive sequence below:
2)
Summation Notation
Sometimes it is necessary to find the sum of many terms of an infinite sequence.
This represents the sum of the first n terms.
Bell Work:
Looking at the following sequence:
7, 16, 25, 34, 43, 52…
What would be the 10th term in this sequence?
What would be the nth term in this sequence?
Find the sum of the 35th through 40th terms.
Arithmetic Sequences
A sequence is considered to be arithmetic if there exists a value d such that d can be added to the previous term to find the next term.
To put it simply:
The value d is called the common difference.
Find the nth term…
When looking at a sequence, we can easily determine the nth term by finding the common difference.
Examples:
1) The first term of an arithmetic sequence is 20, and the sixth term is -10. Find the nth term and then find the 20th term.
2) If the 4th term of a sequence is 5 and the 9th term is 20, find the nth term and then find the 50th term.
Bell Work:
For the arithmetic sequences below, find the 5th, 25th, 100th, and nth terms.
1) -12, -7, -2, 3…
2) 12.5, 9.2, 5.9, 2.6…
3) If the 8th term of a sequence is 32 and the 12th term is -16, find the nth term and then find the 50th term.
Partial Sums
Suppose is an arithmetic sequence with a common difference d, then we can find the partial sum by using the first n terms.
or
Examples:
Find the sum of every even integer from 2 through 300.
Find the sum of every other odd integer from 1 through 513.
Inserting Arithmetic Means
Its exactly like it sounds! We are finding averages between values.
We are trying to find values that are equidistant from each other, so we have to find the common difference!!!
Examples:
Ex: Insert three arithmetic means between 2 and 10.
Ex: Insert seven arithmetic means between 18 and 24.
Bell Work:
Consider the following recursive sequence:
1) If the first term in the sequence is given as 10, what will happen to the terms of the sequence as k gets much larger?
2) What would happen if the first term was 10,000?
Bell Work:
1) Every day after soccer practice, Bobby loses 15% of the sodium in his body through perspiration . With the dinner he eats after practice, he intakes 40 mg of sodium. Write a recursive sequence to show the amount of sodium in Bobby’s system after any given day.
2) If Bobby initially has 400 mg of sodium in his system, how long will it take to drop below 350 mg?
3) If Bobby wants to maintain a level of 380 mg of sodium in his system, how many mg of sodium should he consume after practice each night?
Class Work/Home Work:
Pages 735 – 737 #’s 14, 24, 42, 52 (review 57 and 58)
Pages 742 – 744 #’s 6, 8, 12, 16, 18, 22, 26, 32 – 44 evens
We will review these on Monday, and take a quiz on Tuesday!!!
Geometric Sequences
A sequence is geometric if there exists a real number r such that the next term in a sequence can be found by multiplying the previous term by r.
, this is called the common ratio.
Find the nth term…
This will allow you to find any term of a geometric sequence.
This can also be used to obtain the common ratio r or the first term
Examples:
Ex1: A geometric sequence has a first term of 8 and a common ratio of -1/2. Find the first five terms and the nth term of the sequence.
Ex2: The 3rd term of a geometric sequence is 5 and the 6th term is -40. Find the nth term and the 11th term of the sequence.
Examples:
Ex: If the third term in a geometric sequence is and the 8th term is , find the nth term and the 5th term of the sequence.
Bell Work:
If the third term in a geometric sequence is and the seventh term in the sequence is -270, then find the nth term and the 10th term.
Partial Sums!!!! Yeah boiiiiiiiiii!
The partial sum of a geometric sequence can be found by:
, where r ≠ 1.
Remember, this is different from an arithmetic sequence!
Example:
Suppose Mr. Kelsey decides to save up to buy a castle. He sets aside 1 cent on the first day, 2 cents on the second, 4 on the third, 8 on the fourth, and so on.
A) How much money will he have to set aside on the 18th day?
B) What is the total amount of money he will have saved up after 25 days???
Sum of an Infinite Geometric Sequence!!!!!
If -1 < r < 1, then an infinite geometric series has the sum of:
Ex: Lets find the sum of the following series:
We can only find sums of infinite sequences if the sequence if the common ratio is between 1 and -1!!!!!
Bell Work:
The yearly depreciation of a certain machine is 20% of its value at the beginning of the year. If the original cost of the machine is $400,000, then what is its value after 15 years?
Find the sum of the following infinite geometric series:
Before we get there…
Factorial!!!!!!!!!!!!
6! is read as 6 factorial.
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
4! = 4 x 3 x 2 x 1 = 24
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320
Combinations:
In order to determine binomial coefficients, we need to know how to find a combination.
C(n,k) =
Ex: Lets find C(5,3)
The Binomial Theorem
What does it do?
It allows us to expand any binomial raised to the nth power!
We know that (a + b)² = a² + 2ab + b², but what about expanding (a + b) to the 10th power???
Here we go…
Warning: What you are about to see can be confusing and downright insane, viewer discretion is advised.
Turn to page 765 in your textbooks and look at the top orange table.
This is…The Binomial Theorem.
Pascal’s Triangle
Get ready to be amazed at Pascal’s Triangle…
It used to be called Kelsey’s Triangle, but I am currently in a legal battle over naming rights.
Remember…
When expanding binomials, you will always have (n + 1) terms.
will have 10 terms
You can use the Binomial Theorem discussed in class, or Pascal’s Triangle. Both will work!!!
Finding a Specific Term:
To find a specific term, we use the following:
The (k + 1) term = for some binomial .
Example: Find the fifth term of
Bell Work:
We have a quiz Tuesday/Wednesday, so try the following review problems:
Pages 751 – 753 #’s 10, 20, 28, 34, 36, 48, 50, 53
Page 769 #’s 10, 12, 24, 28, 34, 36, 40
Brief Intro into Derivatives
Tangent Lines Slope Maximum/Minimum Distance/Velocity/Acceleration Zeroes Limits
Notation:
For a given function , its derivative is given as
It goes the same for any function:
The derivative of g(x) is g’(x).
Constant Rule
The Derivative of a Constant is ZERO!
If f(x) = c, where c is a constant, then f’(x) = 0.
The Sum/Difference Rule:
If f and g are both differentiable functions, then:
If h(x) = f(x) + g(x), then h’(x) = f’(x) + g’(x)
and
If h(x) = f(x) - g(x), then h’(x) = f’(x) - g’(x)
Finding the Derivative of Polynomial Expressions:
Find the derivative of the following expressions:
Ex:
Ex:
Ex:
Bell Work:
The maximum and minimum x values for a function can be found by finding the zeroes for the derivative of the original function.
Find the exact values of the maximums and minimums for the following function by using the derivative:
Example:
Use the derivative of the following function to find the exact values of the maximums and minimums of the original function.
Second/Third/Fourth…Derivatives
Given the function:
Find the first derivative.
Find the second derivative.
Find the third derivative.
Product Rule
If f and g are both differentiable functions, then:
If h(x) = f(x)· g(x),
then h’(x) = f(x)· g’(x) + g(x)· f’(x)
Chain Rule
When finding the derivative of a function that contains a composition, we must use the chain rule.
Combining the Chain Rule…
Sometimes it is necessary that in order to find a derivative, you must use the chain rule with another rule (usually the product or quotient rule).
Ex:
Challenge Problem:
Differentiate
If you don’t enjoy this problem, then you have serious issues that need to be worked out.
Double Bonus Challenge Problem!!!!!!!!!!!!
There really is no bonus, but it’s a Thursday double day DOUBLE CHALLENGE PROBLEM YEAH!!!!!!!!!!!!!!!
Find the derivative of
TRIPLE BONUS CHALLENGE PROBLEM!!!!!!!!!!!!!!!!
OMG! It’s a Thursday Triple Day TRIPLE BONUS CHALLENGE PROBLEM!!!!!!
Find the derivative of
QUADRUPLE BONUS CHALL…OKAY, NOT REALLY.
No homework, review tomorrow and Monday. Test will be on Tuesday or Wednesday next week.
Arithmetic Sequences Geometric Sequences Binomial Expansion (Binomial Theorem or
Pascal’s Triangle) Derivatives of Polynomial Functions